Apply to Aakash iACST Scholarship Test 2024
Access premium articles, webinars, resources to make the best decisions for career, course, exams, scholarships, study abroad and much more with
Plan, Prepare & Make the Best Career Choices
NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions provided here. These solutions are created by expert team at Careers360 keeping in mind the latest syllabus and pattern of CBSE 2023-24. Arithmetic Progression class 10 solutions are designed covering all the topics and concepts comprehensively with step by step solutions of all exercise. After practicing these NCERT solutions for Class 10 students get command and in-depth understanding of the concepts which ultimately lead to score well in the exams.
This Class 10 Maths Chapter 5 covers the type of questions in which succeeding terms are obtained by adding the same number to the preceding terms. NCERT solutions are not only important for Board exams but also foundation for higher class therefore, students can take help from these solutions if they are facing any problems. here in this page students can find detailed arithmetic progression class 10 notes NCERT solutions also they can download class 10 maths chapter 5 pdf.
Apply to Aakash iACST Scholarship Test 2024
Also See,
Arithmetic Progressions (AP) involve sequence terms denoted as a1, a2, a3, ..., an representing a series of integers.
An AP maintains a constant difference between consecutive terms, known as the common difference.
For terms a1, a2, a3, a4, a5, and a6 in an AP, the common difference can be expressed as D = a2 – a1 = a3 – a2 = a4 – a3 = ...
The nth term of an AP is given by the formula: an = a + (n – 1) d.
Where
a = First term of the sequence.
n = Term's position in the sequence.
d = Common difference.
The sum of the first 'n’ terms in an AP is calculated using the formula:
Sn = (n/2) [2a + (n – 1)d]
Where
Sn denotes the sum of the terms.
'n' is the number of terms being summed.
'a' is the first term.
'd' stands for the common difference.
Free download NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions PDF for CBSE Exam.
NCERT Solutions for 5 chapter maths class 10 Arithmetic Progressions Excercise: 5.1
Q1 (i) In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (i) The taxi fare after each km when the fare is for the first km and for each additional .
It is given that
Fare for = Rs. 15
And after that for each additional
Now,
Fare for = Fare of first km + Additional fare for 1 km
= Rs. 15 + 8 = Rs 23
Fare for = Fare of first km + Fare of additional second km + Fare of additional third km
= Rs. 23 + 8= Rs 31
Fare of n km =
( We multiplied by n - 1 because the first km was fixed and for rest, we are adding additional fare.
In this, each subsequent term is obtained by adding a fixed number (8) to the previous term.)
Now, we can clearly see that this is an A.P. with the first term (a) = 15 and common difference (d) = 8
Q1 (ii) In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (ii) The amount of air present in a cylinder when a vacuum pump removes of the air remaining in the cylinder at a time.
It is given that
vacum pump removes of the air remaining in the cylinder at a time
Let us take initial quantity of air = 1
Now, the quantity of air removed in first step = 1/4
Remaining quantity after 1 st step
Similarly, Quantity removed after 2 nd step = Quantity removed in first step Remaining quantity after 1 st step
Now,
Remaining quantity after 2 nd step would be = Remaining quantity after 1 st step - Quantity removed after 2 nd step
Now, we can clearly see that
After the second step the difference between second and first and first and initial step is not the same, hence
the common difference (d) is not the same after every step
Therefore, it is not an AP
Q1 (iii) In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (iii) The cost of digging a well after every meter of digging, when it costs for the first metre and rises by for each subsequent meter.
It is given that
Cost of digging of 1st meter = Rs 150
and
rises by for each subsequent meter
Therefore,
Cost of digging of first 2 meters = cost of digging of first meter + cost of digging additional meter
Cost of digging of first 2 meters = 150 + 50
= Rs 200
Similarly,
Cost of digging of first 3 meters = cost of digging of first 2 meters + cost of digging of additional meter
Cost of digging of first 3 meters = 200 + 50
= Rs 250
We can clearly see that 150, 200,250, ... is in AP with each subsequent term is obtained by adding a fixed number (50) to the previous term.
Therefore, it is an AP with first term (a) = 150 and common difference (d) = 50
Q1 (iv) In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (iv) The amount of money in the account every year, when is deposited at compound interest at per annum .
Amount in the beginning = Rs. 10000
Interest at the end of 1st year at the rate of
is of 10000 =
Therefore, amount at the end of 1st year will be
= 10000 + 800
= 10800
Now,
Interest at the end of 2nd year at rate of
is of 10800 =
Therefore,, amount at the end of 2 nd year
= 10800 + 864 = 11664
Since each subsequent term is not obtained by adding a unique number to the previous term; hence, it is not an AP
Q2 (i) Write first four terms of the AP, when the first term a and the common difference d are given as follows
It is given that
Now,
Therefore, the first four terms of the given series are 10,20,30,40
Q2 (ii) Write first four terms of the AP when the first term a and the common difference d are given as follows:
Answer:
It is given that
Now,
Therefore, the first four terms of the given series are -2,-2,-2,-2
Q2 (iii) Write first four terms of the AP when the first term a and the common difference d are given as follows
It is given that
Now,
Therefore, the first four terms of the given series are 4,1,-2,-5
Q2 (iv) Write first four terms of the AP when the first term a and the common difference d are given as follows
It is given that
Now,
Therefore, the first four terms of the given series are
Q2 (v) Write first four terms of the AP when the first term a and the common difference d are given as follows
It is given that
Now,
Therefore, the first four terms of the given series are -1.25,-1.50,-1.75,-2
Q3 (i) For the following APs, write the first term and the common difference:
Given AP series is
Now, first term of this AP series is 3
Therefore,
First-term of AP series (a) = 3
Now,
And common difference (d) =
Therefore, first term and common difference is 3 and -2 respectively
Q3 (ii) For the following APs, write the first term and the common difference:
Answer:
Given AP series is
Now, the first term of this AP series is -5
Therefore,
First-term of AP series (a) = -5
Now,
And common difference (d) =
Therefore, the first term and the common difference is -5 and 4 respectively
Q3 (iii) For the following APs, write the first term and the common difference:
Given AP series is
Now, the first term of this AP series is
Therefore,
The first term of AP series (a) =
Now,
And common difference (d) =
Therefore, the first term and the common difference is and respectively
Q3 (iv) For the following APs, write the first term and the common difference:
Given AP series is
Now, the first term of this AP series is 0.6
Therefore,
First-term of AP series (a) = 0.6
Now,
And common difference (d) =
Therefore, the first term and the common difference is 0.6 and 1.1 respectively.
Q4 (i) Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
the first term to this series is = 2
Now,
We can clearly see that the difference between terms are not equal
Hence, given series is not an AP
Q4 (ii) Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
first term to this series is = 2
Now,
We can clearly see that the difference between terms are equal and equal to
Hence, given series is in AP
Now, the next three terms are
Therefore, next three terms of given series are
Q4 (iii) Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
the first term to this series is = -1.2
Now,
We can clearly see that the difference between terms are equal and equal to -2
Hence, given series is in AP
Now, the next three terms are
Therefore, next three terms of given series are -9.2,-11.2,-13.2
Q4 (iv) Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
the first term to this series is = -10
Now,
We can clearly see that the difference between terms are equal and equal to 4
Hence, given series is in AP
Now, the next three terms are
Therefore, next three terms of given series are 6,10,14
Q4 (v) Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
the first term to this series is = 3
Now,
We can clearly see that the difference between terms are equal and equal to
Hence, given series is in AP
Now, the next three terms are
Therefore, next three terms of given series are
Q4 (vi) Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
Answer:
Given series is
Now,
the first term to this series is = 0.2
Now,
We can clearly see that the difference between terms are not equal
Hence, given series is not an AP
Q4 (vii) Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
first term to this series is = 0
Now,
We can clearly see that the difference between terms are equal and equal to -4
Hence, given series is in AP
Now, the next three terms are
Therefore, the next three terms of given series are -16,-20,-24
Answer:
Given series is
Now,
the first term to this series is =
Now,
We can clearly see that the difference between terms are equal and equal to 0
Hence, given series is in AP
Now, the next three terms are
Therefore, the next three terms of given series are
Q4 (ix) Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
the first term to this series is = 1
Now,
We can clearly see that the difference between terms are not equal
Hence, given series is not an AP
Q4 (x) Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
the first term to this series is = a
Now,
We can clearly see that the difference between terms are equal and equal to a
Hence, given series is in AP
Now, the next three terms are
Therefore, next three terms of given series are 5a,6a,7a
Q4 (xi) Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
Answer:
Given series is
Now,
the first term to this series is = a
Now,
We can clearly see that the difference between terms are not equal
Hence, given series is not in AP
Q4 (xii) Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
Given series is
We can rewrite it as
Now,
first term to this series is = a
Now,
We can clearly see that difference between terms are equal and equal to
Hence, given series is in AP
Now, the next three terms are
Therefore, next three terms of given series are
That is the next three terms are
Q4 (xiii) Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
Given series is
Now,
the first term to this series is =
Now,
We can clearly see that the difference between terms are not equal
Hence, given series is not in AP
Q4 (xiv) Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
Given series is
we can rewrite it as
Now,
the first term to this series is = 1
Now,
We can clearly see that the difference between terms are not equal
Hence, given series is not in AP
Q4 (xv) Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
Given series is
we can rewrite it as
Now,
the first term to this series is = 1
Now,
We can clearly see that the difference between terms are equal and equal to 24
Hence, given series is in AP
Now, the next three terms are
Therefore, the next three terms of given series are 97,121,145
NCERT Solutions Class 10 Chapter 5 Arithmetic Progressions Excercise: 5.2
a | d | n |
| |
(i) (ii) (iii) (iv) (v) | 7
| 3
0 | 8 10 18
105 |
0
|
(i)
It is given that
Now, we know that
Therefore,
(ii) It is given that
Now, we know that
(iii) It is given that
Now, we know that
Therefore,
(iv) It is given that
Now, we know that
Therefore,
(v) It is given that
Now, we know that
Therefore,
Q2 (i) Choose the correct choice in the following and justify: th term of the AP: is
Given series is
Here,
and
Now, we know that
It is given that
Therefore,
Therefore, th term of the AP: is -77
Hence, Correct answer is (C)
Q2 (ii) Choose the correct choice in the following and justify : 11th term of the AP: is
Given series is
Here,
and
Now, we know that
It is given that
Therefore,
Therefore, 11th term of the AP: is 22
Hence, the Correct answer is (B)
Q3 (i) In the following APs, find the missing terms in the boxes :
Given AP series is
Here,
Now, we know that
Now,
Therefore, the missing term is 14
Q3 (ii) In the following APs, find the missing terms in the boxes:
Given AP series is
Here,
Now,
Now, we know that
Now,
And
Therefore, missing terms are 18 and 8
AP series is 18,13,8,3
Q3 (iii) In the following APs, find the missing terms in the boxes :
Given AP series is
Here,
Now, we know that
Now,
And
Therefore, missing terms are and 8
AP series is
Q3 (iv) In the following APs, find the missing terms in the boxes :
Answer:
Given AP series is
Here,
Now, we know that
Now,
And
And
And
Therefore, missing terms are -2,0,2,4
AP series is -4,-2,0,2,4,6
Q3 (v) In the following APs, find the missing terms in the boxes :
Given AP series is
Here,
Now,
Now, we know that
Now,
And
And
And
Therefore, missing terms are 53,23,8,-7
AP series is 53,38,23,8,-7,-22
Q4 Which term of the AP : is ?
Answer:
Given AP is
Let suppose that nth term of AP is 78
Here,
And
Now, we know that that
Therefore, value of 16th term of given AP is 78
Q5 (i) Find the number of terms in each of the following APs :
Given AP series is
Let's suppose there are n terms in given AP
Then,
And
Now, we know that
Therefore, there are 34 terms in given AP
Q5 (ii) Find the number of terms in each of the following APs :
Given AP series is
suppose there are n terms in given AP
Then,
And
Now, we know that
Therefore, there are 27 terms in given AP
Q6 Check whether is a term of the AP :
Given AP series is
Here,
And
Now,
suppose -150 is nth term of the given AP
Now, we know that
Value of n is not an integer
Therefore, -150 is not a term of AP
Q7 Find the st term of an AP whose th term is and the th term is .
It is given that
th term of an AP is and the th term is
Now,
And
On solving equation (i) and (ii) we will get
Now,
Therefore, 31st terms of given AP is 178
Q8 An AP consists of terms of which rd term is and the last term is . Find the th term.
Answer:
It is given that
AP consists of terms of which rd term is and the last term is
Now,
And
On solving equation (i) and (ii) we will get
Now,
Therefore, 29th term of given AP is 64
Q9 If the rd and the th terms of an AP are and respectively, which term of this AP is zero?
Answer:
It is given that
rd and the th terms of an AP are and respectively
Now,
And
On solving equation (i) and (ii) we will get
Now,
Let nth term of given AP is 0
Then,
Therefore, 5th term of given AP is 0
Q10 The th term of an AP exceeds its th term by . Find the common difference.
Answer:
It is given that
th term of an AP exceeds its th term by
i.e.
Therefore, the common difference of AP is 1
Q11 Which term of the AP : will be more than its th term?
Answer:
Given AP is
Here,
And
Now, let's suppose nth term of given AP is more than its th term
Then,
Therefore, 65th term of given AP is more than its th term
It is given that
Two APs have the same common difference and difference between their th terms is
i.e.
Let common difference of both the AP's is d
Now, difference between 1000th term is
Therefore, difference between 1000th term is 100
Q 13 How many three-digit numbers are divisible by ?
Answer:
We know that the first three digit number divisible by 7 is 105 and last three-digit number divisible by 7 is 994
Therefore,
Let there are n three digit numbers divisible by 7
Now, we know that
Therefore, there are 128 three-digit numbers divisible by 7
Q14 How many multiples of lie between and ?
Answer:
We know that the first number divisible by 4 between 10 to 250 is 12 and last number divisible by 4 is 248
Therefore,
Let there are n numbers divisible by 4
Now, we know that
Therefore, there are 60 numbers between 10 to 250 that are divisible by 4
Q15 For what value of , are the th terms of two APs: and equal?
Given two AP's are
and
Let first term and the common difference of two AP's are a , a' and d , d'
And
Now,
Let nth term of both the AP's are equal
Therefore, the 13th term of both the AP's are equal
Q16 Determine the AP whose third term is and the th term exceeds the th term by .
Answer:
It is given that
3rd term of AP is and the th term exceeds the th term by
i.e.
And
Put the value of d in equation (i) we will get
Now, AP with first term = 4 and common difference = 6 is
4,10,16,22,.....
Q17 Find the th term from the last term of the AP : .
Given AP is
Here,
And
Let suppose there are n terms in the AP
Now, we know that
So, there are 51 terms in the given AP and 20th term from the last will be 32th term from the starting
Therefore,
Therefore, 20th term from the of given AP is 158
It is given that
sum of the th and th terms of an AP is and the sum of the th and th terms is
i.e.
And
On solving equation (i) and (ii) we will get
Therefore,first three of AP with a = -13 and d = 5 is
-13,-8,-3
It is given that
Subba Rao started work at an annual salary of Rs 5000 and received an increment of Rs 200 each year
Therefore,
Let's suppose after n years his salary will be Rs 7000
Now, we know that
Therefore, after 11 years his salary will be Rs 7000
after 11 years, starting from 1995, his salary will reach to 7000, so we have to add 10 in 1995, because these numbers are in years
Thus , 1995+10 = 2005
It is given that
Ramkali saved Rs 5 in the first week of a year and then increased her weekly savings by Rs
Therefore,
after th week, her weekly savings become Rs
Now, we know that
Therefore, after 10 weeks her saving will become Rs 20.75
NCERT Arithmetic Progression Class 10 Solutions Excercise: 5.3
Q1 (i) Find the sum of the following APs: to terms.
Given AP is
to terms
Here,
And
Now, we know that
Therefore, the sum of AP to terms is 245
Q1 (ii) Find the sum of the following APs: to terms.
Given AP is
to terms.
Here,
And
Now, we know that
Therefore, the sum of AP to terms. is -180
Q1 (iii) Find the sum of the following APs: to terms.
Given AP is
to terms..
Here,
And
Now, we know that
Therefore, the sum of AP to terms. is 5505
Q1 (iv) Find the sum of the following APs: to terms.
Given AP is
to terms.
Here,
And
Now, we know that
Therefore, the sum of AP to terms. is
Q2 (i) Find the sums given below :
Given AP is
We first need to find the number of terms
Here,
And
Let suppose there are n terms in the AP
Now, we know that
Now, we know that
Therefore, the sum of AP is
Q2 (ii) Find the sums given below :
Given AP is
We first need to find the number of terms
Here,
And
Let suppose there are n terms in the AP
Now, we know that
Now, we know that
Therefore, the sum of AP is 286
Q2 (iii) Find the sums given below :
Given AP is
We first need to find the number of terms
Here,
And
Let suppose there are n terms in the AP
Now, we know that
Now, we know that
Therefore, the sum of AP is -8930
Q3 (i) In an AP: given , , , find and .
It is given that
Let suppose there are n terms in the AP
Now, we know that
Now, we know that
Therefore, the sum of the given AP is 440
Q3 (ii) In an AP: given , , find and .
It is given that
Now, we know that
Therefore, the sum of given AP is 273
Q3 (iii) In an AP: given find and .
It is given that
Now, we know that
Therefore, the sum of given AP is 246
Q3 (iv) In an AP: given find and
Answer:
It is given that
Now, we know that
On solving equation (i) and (ii) we will get
Now,
Therefore, the value of d and 10th terms is -1 and 8 respectively
Q3 (vi) In an AP: given find and .
It is given that
Now, we know that
n can not be negative so the only the value of n is 5
Now,
Therefore, value of n and nth term is 5 and 34 respectively
Q3 (vii) In an AP: given find and .
It is given that
Now, we know that
Now, we know that
Now, put this value in (i) we will get
Therefore, value of n and d are respectively
Q3 (viii) In an AP: given find and .
It is given that
Now, we know that
Now, we know that
Value of n cannot be negative so the only the value of n is 7
Now, put this value in (i) we will get
a = -8
Therefore, the value of n and a are 7 and -8 respectively
Q3 (ix) In an AP: given find .
It is given that
Now, we know that
Therefore, the value of d is 6
Q3 (x) In an AP: given and there are total terms. Find .
It is given that
Now, we know that
Now, we know that
Therefore, the value of a is 4
Q4 How many terms of the AP: must be taken to give a sum of ?
Given AP is
Here,
And
Now , we know that
Value of n can not be negative so the only the value of n is 12
Therefore, the sum of 12 terms of AP must be taken to give a sum of .
It is given that
Now, we know that
Now, we know that
Now, put this value in (i) we will get
Therefore, value of n and d are respectively
It is given that
Now, we know that
Now, we know that
Therefore, there are 38 terms and their sun is 6973
Q7 Find the sum of first terms of an AP in which and nd term is .
It is given that
Now, we know that
Now, we know that
Therefore, there are 22 terms and their sum is 1661
Q8 Find the sum of first terms of an AP whose second and third terms are and respectively.
It is given that
And
Now,
Now, we know that
Therefore, there are 51 terms and their sum is 5610
Q9 If the sum of first terms of an AP is and that of terms is , find the sum of first terms.
It is given that
Now, we know that
Similarly,
On solving equation (i) and (ii) we will get
a = 1 and d = 2
Now, the sum of first n terms is
Therefore, the sum of n terms is
Q10 (i) Show that form an AP where an is defined as below : Also find the sum of the first terms.
It is given that
We will check values of for different values of n
and so on.
From the above, we can clearly see that this is an AP with the first term(a) equals to 7 and common difference (d) equals to 4
Now, we know that
Therefore, the sum of 15 terms is 525
Q10 (ii) Show that form an AP where an is defined as below : . Also find the sum of the first terms in each case.
It is given that
We will check values of for different values of n
and so on.
From the above, we can clearly see that this is an AP with the first term(a) equals to 4 and common difference (d) equals to -5
Now, we know that
Therefore, the sum of 15 terms is -465
It is given that
the sum of the first terms of an AP is
Now,
Now, first term is
Therefore, first term is 3
Similarly,
Therefore, sum of first two terms is 4
Now, we know that
Now,
Similarly,
Q12 Find the sum of the first positive integers divisible by .
Positive integers divisible by 6 are
6,12,18,...
This is an AP with
Now, we know that
Therefore, sum of the first positive integers divisible by is 4920
Q13 Find the sum of the first multiples of .
First 15 multiples of 8 are
8,16,24,...
This is an AP with
Now, we know that
Therefore, sum of the first 15 multiple of 8 is 960
Q14 Find the sum of the odd numbers between and .
The odd number between 0 and 50 are
1,3,5,...49
This is an AP with
There are total 25 odd number between 0 and 50
Now, we know that
Therefore, sum of the odd numbers between and 625
It is given that
Penalty for delay of completion beyond a certain date is Rs for the first day, Rs for the second day, Rs for the third day and penalty for each succeeding day being Rs more than for the preceding day
We can clearly see that
200,250,300,..... is an AP with
Now, the penalty for 30 days is given by the expression
Therefore, the penalty for 30 days is 27750
It is given that
Each price is decreased by 20 rupees,
Therefore, d = -20 and there are total 7 prizes so n = 7 and sum of prize money is Rs 700 so
Let a be the prize money given to the 1st student
Then,
Therefore, the prize given to the first student is Rs 160
Now,
Let is the prize money given to the next 6 students
then,
Therefore, prize money given to 1 to 7 student is 160,140,120,100,80,60.40
First there are 12 classes and each class has 3 sections
Since each section of class 1 will plant 1 tree, so 3 trees will be planted by 3 sections of class 1. Thus every class will plant 3 times the number of their class
Similarly,
No. of trees planted by 3 sections of class 1 = 3
No. of trees planted by 3 sections of class 2 = 6
No. of trees planted by 3 sections of class 3 = 9
No. of trees planted by 3 sections of class 4 = 12
Its clearly an AP with first term (a) = 3 and common difference (d) = 3 and total number of classes (n) = 12
Now, number of trees planted by 12 classes is given by
Therefore, number of trees planted by 12 classes is 234
[ Hint : Length of successive semicircles is with centres at respectively.]
From the above-given figure
Circumference of 1st semicircle
Similarly,
Circumference of 2nd semicircle
Circumference of 3rd semicircle
It is clear that this is an AP with
Now, sum of length of 13 such semicircles is given by
Therefore, sum of length of 13 such semicircles is 143 cm
As the rows are going up, the no of logs are decreasing,
We can clearly see that 20, 19, 18, ..., is an AP.
and here
Let suppose 200 logs are arranged in 'n' rows,
Then,
Now,
case (i) n = 25
But number of rows can not be in negative numbers
Therefore, we will reject the value n = 25
case (ii) n = 16
Therefore, the number of rows in which 200 logs are arranged is equal to 5
Distance travelled by the competitor in picking and dropping 1st potato
Distance travelled by the competitor in picking and dropping 2nd potato
Distance travelled by the competitor in picking and dropping 3rd potato
and so on
we can clearly see that it is an AP with first term (a) = 10 and common difference (d) = 6
There are 10 potatoes in the line
Therefore, total distance travelled by the competitor in picking and dropping potatoes is
Therefore, the total distance travelled by the competitor in picking and dropping potatoes is 370 m
NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions Excercise: 5.4
Q1 Which term of the AP: is its first negative term? [ Hint : Find for ]
Given AP is
Here
Let suppose nth term of the AP is first negative term
Then,
If nth term is negative then
Therefore, first negative term must be 32nd term
It is given that sum of third and seventh terms of an AP are and their product is
Now,
And
put value from equation (i) in (ii) we will get
Now,
case (i)
Then,
case (ii)
Then,
Q3 A ladder has rungs cm apart. (see Fig. ). The rungs decrease uniformly in length from cm at the bottom to cm at the top. If the top and the bottom rungs are m apart, what is the length of the wood required for the rungs? [ Hint: Number of rungs
It is given that
The total distance between the top and bottom rung
Distance between any two rungs = 25 cm
Total number of rungs =
And it is also given that bottom-most rungs is of 45 cm length and topmost is of 25 cm length.As it is given that the length of rungs decrease uniformly, it will form an AP with
Now, we know that
Now, total length of the wood required for the rungs is equal to
Therefore, the total length of the wood required for the rungs is equal to 385 cm
It is given that the sum of the numbers of the houses preceding the house numbered is equal to the sum of the numbers of the houses following it
And 1,2,3,.....,49 form an AP with a = 1 and d = 1
Now, we know that
Suppose their exist an n term such that ( n < 49)
Now, according to given conditions
Sum of first n - 1 terms of AP = Sum of terms following the nth term
Sum of first n - 1 term of AP = Sum of whole AP - Sum of first m terms of AP
i.e.
Given House number are not negative so we reject n = -35
Therefore, the sum of no of houses preceding the house no 35 is equal to the sum of no of houses following the house no 35
It is given that
football ground comprises of steps each of which is m long and Each step has a rise of and a tread of
Now,
The volume required to make the first step =
Similarly,
The volume required to make 2nd step =
And
The volume required to make 3rd step =
And so on
We can clearly see that this is an AP with
Now, the total volume of concrete required to build the terrace of 15 such step is
Therefore, the total volume of concrete required to build the terrace of 15 such steps is
Checking whether the series is an arithmetic progression or not.
To find the nth term or last term or an arithmetic progression.
Questions based on the sum of an arithmetic progression
Use of linear equation concept in an arithmetic progression
Application of nth term and sum formula
NCERT Class 10 Arithmetic Progression Exercise-Wise Solutions
The NCERT solutions cover all the questions given in the exercise and in between the chapters. This makes it really helpful in homework as well as in the board exam preparations.
The Arithmetic Progression Class 10 Maths Chapter 5 NCERT solutions offer a step by step description of the solutions.
These NCERT solutions for Class 10 Maths ch 5 Arithmetic Progressions is given in a simple language and are easy to understand.
For NCERT Solutions for Class 10 Maths Chapter 5 PDF download, you can save the webpage of this article.
Chapter No. | Chapter Name |
Chapter 1 | |
Chapter 2 | |
Chapter 3 | |
Chapter 4 | |
Chapter 5 | Arithmetic Progressions |
Chapter 6 | |
Chapter 7 | |
Chapter 8 | |
Chapter 9 | |
Chapter 10 | |
Chapter 11 | |
Chapter 12 | |
Chapter 13 | |
Chapter 14 | |
Chapter 15 |
Before practising the NCERT chapter 5 Maths Class 10 solutions, go through the conceptual theory given in the NCERT textbook.
After going through the conceptual part, jump on to practice exercises available.
While solving the exercises if you are facing problems in any specific question, then take the help of NCERT solutions for Class 10 Maths chapter 5 Arithmetic Progressions.
Once you have done the practice exercises you can move to previous year questions.
Checking whether the series is an arithmetic progression or not, nth term or last term of arithmetic progression, sum of an arithmetic progression, use of linear equation concept in an arithmetic progression and application of nth term and sum of a.p. formulae are the important topics in this chapter.
The NCERT Solutions for Class 10 Maths Chapter 5 includes a range of question types such as multiple choice, long and short answer, fill in the blank, and real-world examples. These questions aim to enhance students' problem-solving and time management abilities, thus preparing them for success in their CBSE exams.
The number of exercises in Maths Chapter 5 for Class 10 are four. These are very important because these include all types of questions on which repeatedly questions are being asked in board exams. Students can practice and find solutions at careers360 official website
It is generally a good idea to practice all of the questions provided in the NCERT solutions for class 10 maths arithmetic progression, as they are designed to cover the key concepts and skills that are important for understanding the chapter and doing well in your exams. By practicing all the questions and arithmetic progression solutions, you will be able to identify your strengths and weaknesses and work on them accordingly. Also, chapter 5 questions and ap solutions class 10 help you build your problem-solving skills, time management skills and confidence, which will be beneficial for you in your exams. Students can also use arithmetic progression class 10 pdf as to study offline.
There are several reasons why you should practice the class 10 maths NCERT solution arithmetic progression. Practising the NCERT Solutions for Class 10 Maths Chapter 5 is a valuable step in your studies as it will help you to understand the concepts, improve your problem-solving skills, and prepare for your exams. Students can use NCERT solutions for class 10 maths arithmetic progression pdf to study comfortable this chapter.
Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. In other words, it is a sequence of numbers in which the difference between two consecutive terms is a constant. Arithmetic progression class 10 NCERT solutions pdf covers various concepts related to arithmetic progressions such as finding the common difference, the nth term, the sum of an arithmetic series, and solving word problems using arithmetic progressions.
Dear aspirant !
Hope you are doing well ! The class 10 Hindi mp board sample paper can be found on the given link below . Set your time and give your best in the sample paper it will lead to great results ,many students are already doing so .
Hope you get it !
Thanking you
The eligibility age criteria for class 10th CBSE is 14 years of age. Since your son will be 15 years of age in 2024, he will be eligible to give the exam.
According to CBSE norms, a student must be 14 years old by the end of the year in which the exam will be held in order to sit for the 10th board exam. If your age is greater than 14, however, there are no limits. Therefore, based on your DOB, you will be 12 years and 6 months old by the end of December 2022. After the actual 16 June 2024, you'll be qualified to take your 10th board.
Hello aspirant,
Central Board Of Secondary Education(CBSE) is likely to declare class 10 and 12 terms 2 board result 2022 by July 15. The evaluation process is underway. Students are demanding good results and don't want to lack behind. They are requesting the board to use their best scores in Term 1 and Term 2 exams to prepare for the results.
CBSE concluded board exams 2022 for 10, and 12 on June 15 and May 24. Exams for both classes began on April 26. A total of 35 lakh students including 21 lakh class 10 students and 14 lakh class 12 students appeared in exams and are awaiting their results.
You can look for your results on websites- cbse.gov (//cbse.gov) .in, cbseresults.nic.in
Thank you
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.
The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.
How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
An investment director is a person who helps corporations and individuals manage their finances. They can help them develop a strategy to achieve their goals, including paying off debts and investing in the future. In addition, he or she can help individuals make informed decisions.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.
Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.
Urban Planning careers revolve around the idea of developing a plan to use the land optimally, without affecting the environment. Urban planning jobs are offered to those candidates who are skilled in making the right use of land to distribute the growing population, to create various communities.
Urban planning careers come with the opportunity to make changes to the existing cities and towns. They identify various community needs and make short and long-term plans accordingly.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
The hospital Administrator is in charge of organising and supervising the daily operations of medical services and facilities. This organising includes managing of organisation’s staff and its members in service, budgets, service reports, departmental reporting and taking reminders of patient care and services.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).
Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
As per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters
Get up to 90% scholarship on NEET, JEE & Foundation courses
Enroll in Resonance Coaching for success in JEE/NEET exams
Start your JEE preparation with ALLEN
Just Study 32% of the NEET syllabus and Score upto 100% marks
As per latest 2024 syllabus. Study 40% syllabus and score upto 100% marks in JEE