# NCERT Solutions For Class 10 Maths Chapter 5 Arithmetic Progressions

NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions: We can see many patterns in nature such as the arrangement of petals of a flower, the pattern of honeycomb, etc. Solutions of NCERT class 10 maths chapter 5 Arithmetic Progressions will give you assistance while doing homework as well as while preparing for the examinations by using such practical examples. We follow certain patterns in daily life. For example, Seema puts rupees 1000 into her daughter’s money box when she was one year old and increased the amount by 100 every year. Then the pattern will be 1000,1100,1200,...........for 1st, 2nd, 3rd,................years. In CBSE NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions, we will explain such type of questions in which succeeding terms are obtained by adding the same number to the preceding terms. In NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions we will also study how to find their general term and the sum of n consecutive terms in the pattern mentioned above, and usage of this knowledge in solving some daily life problems. You can download the NCERT solutions by clicking on the link.

The sum of first n terms is given by

$S_n=\frac{n}{2}(2a+(n-1)d)=\frac{n}{2}(first\ term+last\ term)$

where a is the first term, n is the number of terms in the given A.P and d is a common difference.

## Types of questions asked from class 10 maths chapter 5 Arithmetic Progression

• 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 formulae

## NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions Excercise: 5.1

It is given that
Fare for  $1^{st} \ km$ = Rs. 15
And after that $\small Rs\hspace{1mm}8$  for each additional  $\small km$
Now,
Fare for  $2^{nd} \ km$ = Fare of first km + Additional fare for 1 km
=  Rs. 15 + 8 = Rs 23

Fare for $3^{rd} \ km$ = Fare of first km + Fare of additional second km + Fare of additional third km
= Rs. 23 + 8= Rs 31

Fare of n km =  $15 + 8 \times (n - 1)$
( 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

It is given that
vacum pump removes   $\small \frac{1}{4}$  of the air  remaining in the cylinder at a time
Let us take initial quantity of air = 1

Now, quantity of air removed in first step = 1/4

Remaining quantity after 1st step

$= 1-\frac{1}{4}= \frac{3}{4}$

Similarly, Quantity removed after 2nd step = Quantity removed in first step $\times$  Remaining quantity after 1st step

$=\frac{3}{4}\times \frac{1}{4}= \frac{3}{16}$
Now,

Remaining quantity after 2nd step would be = Remaining quantity after 1st step - Quantity removed after 2nd step

$=\frac{3}{4}- \frac{3}{16}= \frac{12-3}{16}= \frac{9}{16}$
Now, we can clearly see that

After second step the difference between second and first and first and initial step is not the same, hence

the common difference (d)  is not same after every step

Therefore, it is not an AP

It is given that
Cost of digging of 1st meter = Rs 150
and
rises by  $\small Rs\hspace{1mm}50$  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

Amount in the beginning = Rs. 10000

Interest at the end of 1st year at the rate of $\small 8\hspace{1mm}\%$
is $\small 8\hspace{1mm}\%$ of 10000 = $\frac{8\times 10000}{100}= 800$

Therefore, amount at the end of 1st year will be

= 10000 + 800

= 10800

Now,

Interest at the end of 2nd year at rate of $\small 8\hspace{1mm}\%$
is  $\small 8\hspace{1mm}\%$ of 10800 =$\frac{8\times 10800}{100}= 864$

Therefore,, amount at the end of 2ndyear

= 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

It is given that
$\small a=10,d=10$
Now,
$a_1= a =10$
$a_2= a_1 + d= 10 + 10 = 20$
$a_3= a_2 + d= 20 + 10 = 30$
$a_4= a_3 + d= 30 + 10 = 40$
Therefore, the first four terms of the given series are 10,20,30,40

## Q2 Write first four terms of the AP when the first term a and the common difference d are given as follows: (ii) $\small a=-2,d=0$Answer:

It is given that
$\small a=-2,d=0$
Now,
$a_1= a = -2$
$a_2= a_1 + d= -2 + 0 = -2$
$a_3= a_2 + d= -2 + 0 = -2$
$a_4= a_3 + d= -2 + 0 = -2$
Therefore, the first four terms of the given series are -2,-2,-2,-2

It is given that
$\small a=4,d=-3$
Now,
$a_1= a =4$
$a_2= a_1 + d= 4 - 3 = 1$
$a_3= a_2 + d= 1 - 3 = -2$
$a_4= a_3 + d= -2- 3 = -5$
Therefore, the first four terms of the given series are 4,1,-2,-5

It is given that
$\small a=-1,d=\frac{1}{2}$
Now,
$a_1= a =-1$
$a_2= a_1 + d= -1 + \frac{1}{2} = -\frac{1}{2}$
$a_3= a_2 + d= -\frac{1}{2} + \frac{1}{2} = 0$
$a_4= a_3 + d= 0+\frac{1}{2}= \frac{1}{2}$
Therefore, the first four terms of the given series are  $-1,-\frac{1}{2},0, \frac{1}{2}$

It is given that
$\small a=-1.25,d=-0.25$
Now,
$a_1= a =-1.25$
$a_2= a_1 + d= -1.25 -0.25= -1.50$
$a_3= a_2 + d= -1.50-0.25=-1.75$
$a_4= a_3 + d= -1.75-0.25=-2$
Therefore, the first four terms of the given series are  -1.25,-1.50,-1.75,-2

Given AP series is

$\small 3,1,-1,-3,...$

Now, first term of this AP series is 3

Therefore,

First-term of AP series (a) = 3

Now,

$a_1=3 \ \ and \ \ a_2 = 1$

And common difference (d)  = $a_2-a_1 = 1-3 = -2$

Therefore, first term and common difference is 3 and -2 respectively

Given AP series is

$\small -5,-1,3,7,...$

Now, the first term of this AP series is -5

Therefore,

First-term of AP series (a) = -5

Now,

$a_1=-5 \ \ and \ \ a_2 = -1$

And common difference (d)  = $a_2-a_1 = -1-(-5) = 4$

Therefore, the first term and the common difference is -5 and  4 respectively

Given AP series is

$\small \frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{13}{3},...$
Now, the first term of this AP series is $\frac{1}{3}$

Therefore,

The first term of AP series (a) = $\frac{1}{3}$

Now,

$a_1=\frac{1}{3} \ \ and \ \ a_2 = \frac{5}{3}$

And common difference (d)  = $a_2-a_1 = \frac{5}{3}-\frac{1}{3} = \frac{5-1}{3} =\frac{4}{3}$

Therefore, the first term and the common difference is $\frac{1}{3}$ and  $\frac{4}{3}$ respectively

Given AP series is

$\small 0.6,1.7,2.8,3.9,...$

Now, the first term of this AP series is 0.6

Therefore,

First-term of AP series (a) = 0.6

Now,

$a_1=0.6 \ \ and \ \ a_2 = 1.7$

And common difference (d)  = $a_2-a_1 = 1.7-0.6 = 1.1$

Therefore, the first term and the common difference is 0.6 and  1.1 respectively.

Given series is
$\small 2,4,8,12,...$
Now,
the first term to this series is = 2
Now,
$a_1 = 2 \ \ and \ \ a_2 = 4 \ \ and \ \ a_3 = 8$
$a_2-a_1 = 4-2 = 2$
$a_3-a_2 = 8-4 = 4$
We can clearly see that the difference between terms are not equal
Hence, given series is not an AP

Given series is
$\small 2,\frac{5}{2},3,\frac{7}{2},...$
Now,
first term to this series is = 2
Now,
$a_1 = 2 \ \ and \ \ a_2 = \frac{5}{2} \ \ and \ \ a_3 = 3 \ \ and \ \ a_4 = \frac{7}{2}$
$a_2-a_1 = \frac{5}{2}-2 = \frac{5-4}{2}=\frac{1}{2}$
$a_3-a_2 = 3-\frac{5}{2} = \frac{6-5}{2} = \frac{1}{2}$
$a_4-a_3=\frac{7}{2}-3=\frac{7-6}{2} =\frac{1}{2}$
We can clearly see that the difference between terms are equal  and equal to $\frac{1}{2}$
Hence, given series is in AP
Now, the next three terms are
$a_5=a_4+d = \frac{7}{2}+\frac{1}{2} = \frac{8}{2}=4$
$a_6=a_5+d = 4+\frac{1}{2} = \frac{8+1}{2}=\frac{9}{2}$
$a_7=a_6+d =\frac{9}{2} +\frac{1}{2} = \frac{10}{2}=5$

Therefore, next three terms of given series are  $4,\frac{9}{2} ,5$

Given series is
$\small -1.2,-3.2,-5.2,-7.2,...$
Now,
the first term to this series is = -1.2
Now,
$a_1 = -1.2 \ \ and \ \ a_2 = -3.2 \ \ and \ \ a_3 = -5.2 \ \ and \ \ a_4 = -7.2$
$a_2-a_1 = -3.2-(-1.2) =-3.2+1.2=-2$
$a_3-a_2 = -5.2-(-3.2) =-5.2+3.2 = -2$
$a_4-a_3=-7.2-(-5.2)=-7.2+5.2=-2$
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
$a_5=a_4+d = -7.2-2 =-9.2$
$a_6=a_5+d = -9.2-2 =-11.2$
$a_7=a_6+d = -11.2-2 =-13.2$
Therefore, next three terms of given series are  -9.2,-11.2,-13.2

Given series is
$\small -10,-6,-2,2,...$
Now,
the first term to this series is = -10
Now,
$a_1 = -10 \ \ and \ \ a_2 = -6 \ \ and \ \ a_3 = -2 \ \ and \ \ a_4 = 2$
$a_2-a_1 = -6-(-10) =-6+10=4$
$a_3-a_2 = -2-(-6) =-2+6 = 4$
$a_4-a_3=2-(-2)=2+2=4$
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
$a_5=a_4+d = 2+4 =6$
$a_6=a_5+d = 6+4=10$
$a_7=a_6+d = 10+4=14$
Therefore, next three terms of given series are  6,10,14

Given series is
$\small 3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2},...$
Now,
the first term to this series is = 3
Now,
$a_1 = 3 \ \ and \ \ a_2 = 3+\sqrt2 \ \ and \ \ a_3 = 3+2\sqrt2 \ \ and \ \ a_4 = 3+3\sqrt2$
$a_2-a_1 = 3+\sqrt2-3= \sqrt2$
$a_3-a_2 = 3+2\sqrt2-3-\sqrt2 = \sqrt2$
$a_4-a_3 = 3+3\sqrt2-3-2\sqrt2 = \sqrt2$
We can clearly see that the difference between terms are equal  and equal to $\sqrt2$
Hence, given series is in AP
Now, the next three terms are
$a_5=a_4+d = 3+3\sqrt2+\sqrt2=3+4\sqrt2$
$a_6=a_5+d = 3+4\sqrt2+\sqrt2=3+5\sqrt2$
$a_7=a_6+d = 3+5\sqrt2+\sqrt2=3+6\sqrt2$
Therefore, next three terms of given series are  $3+4\sqrt2, 3+5\sqrt2,3+6\sqrt2$

Given series is
$\small 0.2,0.22,0.222,0.2222,...$
Now,
the first term to this series is = 0.2
Now,
$a_1 = 0.2 \ \ and \ \ a_2 = 0.22 \ \ and \ \ a_3 = 0.222 \ \ and \ \ a_4 = 0.2222$
$a_2-a_1 = 0.22-0.2=0.02$
$a_3-a_2 = 0.222-0.22=0.002$

We can clearly see that the difference between terms are not equal
Hence, given series is not an AP

Given series is
$\small 0,-4,-8,-12,...$
Now,
first term to this series is = 0
Now,
$a_1 = 0 \ \ and \ \ a_2 = -4 \ \ and \ \ a_3 = -8 \ \ and \ \ a_4 = -12$
$a_2-a_1 = -4-0 =-4$
$a_3-a_2 = -8-(-4) =-8+4 = -4$
$a_4-a_3=-12-(-8)=-12+8=-4$
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
$a_5=a_4+d = -12-4 =-16$
$a_6=a_5+d = -16-4=-20$
$a_7=a_6+d = -20-4=-24$
Therefore, the next three terms of given series are  -16,-20,-24

Given series is
$\small -\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},...$
Now,
the first term to this series is = $-\frac{1}{2}$
Now,
$a_1 = -\frac{1}{2} \ \ and \ \ a_2 = -\frac{1}{2} \ \ and \ \ a_3 = -\frac{1}{2} \ \ and \ \ a_4 = -\frac{1}{2}$
$a_2-a_1 = -\frac{1}{2}-\left ( -\frac{1}{2} \right ) = -\frac{1}{2}+\frac{1}{2}=0$
$a_3-a_2 = -\frac{1}{2}-\left ( -\frac{1}{2} \right ) = -\frac{1}{2}+\frac{1}{2}=0$
$a_4-a_3 = -\frac{1}{2}-\left ( -\frac{1}{2} \right ) = -\frac{1}{2}+\frac{1}{2}=0$
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
$a_5=a_4+d = -\frac{1}{2}+0=-\frac{1}{2}$
$a_6=a_5+d = -\frac{1}{2}+0=-\frac{1}{2}$
$a_7=a_6+d = -\frac{1}{2}+0=-\frac{1}{2}$
Therefore, the next three terms of given series are  $-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}$

Given series is
$\small 1,3,9,27,...$
Now,
the first term to this series is = 1
Now,
$a_1 = 1 \ \ and \ \ a_2 = 3 \ \ and \ \ a_3 = 9 \ \ and \ \ a_4 = 27$
$a_2-a_1 = 3-1=2$
$a_3-a_2 =9-3=6$

We can clearly see that the difference between terms are not  equal
Hence, given series is not an AP

Given series is
$\small a,2a,3a,4a,...$
Now,
the first term to this series is = a
Now,
$a_1 = a \ \ and \ \ a_2 = 2a \ \ and \ \ a_3 = 3a \ \ and \ \ a_4 = 4a$
$a_2-a_1 = 2a-a =a$
$a_3-a_2 = 3a-2a =a$
$a_4-a_3=4a-3a=a$
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
$a_5=a_4+d =4a+a=5a$
$a_6=a_5+d =5a+a=6a$
$a_7=a_6+d =6a+a=7a$
Therefore, next three terms of given series are  5a,6a,7a

Given series is
$\small a,a^2,a^3,a^4,...$
Now,
the first term to this series is = a
Now,
$a_1 = a \ \ and \ \ a_2 = a^2 \ \ and \ \ a_3 = a^3 \ \ and \ \ a_4 = a^4$
$a_2-a_1 = a^2-a =a(a-1)$
$a_3-a_2 = a^3-a^2 =a^2(a-1)$

We can clearly see that the difference between terms are not equal
Hence, given series is not in AP

Given series is
$\small \sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},...$
We can rewrite it as
$\sqrt2,2\sqrt2,3\sqrt2,4\sqrt2,....$
Now,
first term to this series is = a
Now,
$a_1 = \sqrt2 \ \ and \ \ a_2 = 2\sqrt2 \ \ and \ \ a_3 = 3\sqrt2 \ \ and \ \ a_4 = 4\sqrt2$
$a_2-a_1 = 2\sqrt2-\sqrt2 =\sqrt2$
$a_3-a_2 = 3\sqrt2-2\sqrt2 =\sqrt2$
$a_4-a_3=4\sqrt2-3\sqrt2=\sqrt2$
We can clearly see that difference between terms are equal  and equal to $\sqrt2$
Hence, given series is in AP
Now, the next three terms are
$a_5=a_4+d =4\sqrt2+\sqrt2=5\sqrt2$
$a_6=a_5+d =5\sqrt2+\sqrt2=6\sqrt2$
$a_7=a_6+d =6\sqrt2+\sqrt2=7\sqrt2$
Therefore, next three terms of given series are $5\sqrt2,6\sqrt2,7\sqrt2$

That is the next three terms are $\sqrt{50},\ \sqrt{72},\ \sqrt{98}$

Given series is
$\small \sqrt{3},\sqrt{6},\sqrt{9},\sqrt{12},...$
Now,
the first term to this series is = $\sqrt3$
Now,
$a_1 = \sqrt3 \ \ and \ \ a_2 = \sqrt6 \ \ and \ \ a_3 = \sqrt9 \ \ and \ \ a_4 = \sqrt{12}$
$a_2-a_1 = \sqrt6-\sqrt3 =\sqrt3(\sqrt2-1)$
$a_3-a_2 = 3-\sqrt3 =\sqrt3(\sqrt3-1)$
We can clearly see that the difference between terms are not equal
Hence, given series is not in AP

Given series is
$\small 1^2,3^2,5^2,7^2,...$
we can rewrite it as
$1,9,25,49,....$
Now,
the first term to this series is = 1
Now,
$a_1 =1 \ \ and \ \ a_2 = 9 \ \ and \ \ a_3 =25 \ \ and \ \ a_4 = 49$
$a_2-a_1 = 9-1 = 8$
$a_3-a_2 = 25-9=16$
We can clearly see that the difference between terms are not equal
Hence, given series is not in AP

Given series is
$\small 1^2,5^2,7^2,73,...$
we can rewrite it as
$1,25,49,73....$
Now,
the first term to this series is = 1
Now,
$a_1 =1 \ \ and \ \ a_2 = 25 \ \ and \ \ a_3 =49 \ \ and \ \ a_4 = 73$
$a_2-a_1 = 25-1 = 24$
$a_3-a_2 = 49-25=24$
$a_4-a_3 = 73-49=24$
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
$a_5=a_4+d = 73+24=97$
$a_6=a_5+d = 97+24=121$
$a_7=a_6+d = 121+24=145$
Therefore, the next three terms of given series are 97,121,145

NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions Excercise: 5.2

 a d n $\small a_n$ (i) (ii) (iii) (iv) (v) 7 $\small -18$ $\small ...$ $\small -18.9$ $\small 3.5$ 3 $\small ...$ $\small -3$ $\small 2.5$ 0 8 10 18 $\small ...$ 105 $\small ...$ 0 $\small -5$ $\small 3.6$ $\small ...$

(i)
It is given that
$a=7, d = 3 , n = 8$
Now,  we know that
$a_n = a+(n-1)d$
$a_8 = 7+(8-1)3= 7+7\times 3 = 7+21 = 28$
Therefore,
$a_8 = 28$

(ii) It is given that
$a=-18, n = 10, a_{10} = 0$
Now,  we know that
$a_n = a+(n-1)d$
$a_{10} = -18+(10-1)d$
$0 +18=9d$
$d = \frac{18}{9}=2$
(iii) It is given that
$d=-3, n = 18, a_{18} = -5$
Now,  we know that
$a_n = a+(n-1)d$
$a_{18} = a+(18-1)(-3)$
$-5=a+17\times (-3)$
$a = 51-5 = 46$
Therefore,
$a = 46$

(iv) It is given that
$a=-18.9, d = 2.5, a_{n} = 3.6$
Now,  we know that
$a_n = a+(n-1)d$
$a_{n} = -18.9+(n-1)2.5$
$3.6+18.9= 2.5n-2.5$
$n = \frac{22.5+2.5}{2.5}= \frac{25}{2.5}= 10$
Therefore,
$n = 10$

(v) It is given that
$a=3.5, d = 0, n = 105$
Now,  we know that
$a_n = a+(n-1)d$
$a_{105} = 3.5+(105-1)0$
$a_{105} = 3.5$
Therefore,
$a_{105} = 3.5$

(A) $\small 97$             (B) $\small 77$             (C) $\small -77$             (D) $\small -87$

Given series is
$\small 10,7,4,...,$
Here, $a = 10$
and
$d = 7 - 10 = -3$
Now, we know that
$a_n = a+(n-1)d$
It is given that $n = 30$
Therefore,
$a_{30} = 10+(30-1)(-3)$
$a_{30} = 10+(29)(-3)$
$a_{30} = 10-87 = -77$
Therefore, $\small 30$th term of the AP:  $\small 10,7,4,...,$   is -77

(A) $\small 28$              (B) $\small 22$              (C)  $\small -38$             (D)  $\small -48\frac{1}{2}$

Given series is
$\small -3,-\frac{1}{2},2,...,$
Here, $a = -3$
and
$d =-\frac{1}{2} -(-3)= -\frac{1}{2} + 3 = \frac{-1+6}{2}= \frac{5}{2}$
Now, we know that
$a_n = a+(n-1)d$
It is given that $n = 11$
Therefore,
$a_{11} = -3+(11-1)\left ( \frac{5}{2} \right )$
$a_{11} = -3+(10)\left ( \frac{5}{2} \right )$
$a_{11} = -3+5\times 5 = -3+25 = 22$
Therefore, 11th term of the AP: $\small -3,-\frac{1}{2},2,...,$    is 22

Given AP series is
$\small 2,\hspace {1mm}\fbox{ },\hspace {1mm} 26$
Here, $a = 2 , n = 3 \ and \ a_3 = 26$
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow a_3 =2+(3-1)d$
$\Rightarrow 26 -2=(2)d$
$\Rightarrow d = \frac{24}{2}= 12$
Now,
$a_2= a_1+d$
$a_2= 2+12 = 14$
Therefore, the missing term is 14

(ii)   $\small \fbox { },\hspace {1mm}13,\hspace{1mm}\fbox { }, \hspace {1mm} 3$

Given AP series is
$\small \fbox { },\hspace {1mm}13,\hspace{1mm}\fbox { }, \hspace {1mm} 3$
Here, $a_2 = 13 , n = 4 \ and \ a_4 = 3$
Now,
$a_2= a_1+d$
$a_1= a = 13 - d$
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow a_4 =13-d+(4-1)d$
$\Rightarrow 3-13=-d+3d$
$\Rightarrow d = -\frac{10}{2}= -5$
Now,
$a_2= a_1+d$
$a_1= a = 13 - d= 13-(-5 ) = 18$
And
$a_3=a_2+d$
$a_3=13-5 = 8$
Therefore, missing terms are  18 and 8
AP series is 18,13,8,3

Given AP series is
$\small 5,\: \fbox { },\: \fbox { },\: 9\frac{1}{2}$
Here, $a = 5 , n = 4 \ and \ a_4 = 9\frac{1}{2}= \frac{19}{2}$
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow a_4 =5+(4-1)d$
$\Rightarrow \frac{19}{2} -5=3d$
$\Rightarrow d = \frac{19-10}{2\times 3} = \frac{9}{6} = \frac{3}{2}$
Now,
$a_2= a_1+d$
$a_2 = 5+\frac{3}{2} = \frac{13}{2}$
And
$a_3=a_2+d$
$a_3=\frac{13}{2}+\frac{3}{2} = \frac{16}{2} = 8$
Therefore, missing terms are  $\frac{13}{2}$ and 8
AP series is $5,\frac{13}{2}, 8 , \frac{19}{2}$

## Q3 In the following APs, find the missing terms in the boxes : (iv)   $\small -4,\: \fbox { },\: \fbox { },\: \fbox { },\: \fbox { },\: 6$

Given AP series is
$\small -4,\: \fbox { },\: \fbox { },\: \fbox { },\: \fbox { },\: 6$
Here, $a = -4 , n = 6 \ and \ a_6 = 6$
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow a_6 =-4+(6-1)d$
$\Rightarrow 6+4 = 5d$
$\Rightarrow d = \frac{10}{5} = 2$
Now,
$a_2= a_1+d$
$a_2 = -4+2 = -2$
And
$a_3=a_2+d$
$a_3=-2+2 = 0$
And
$a_4 = a_3+d$
$a_4 = 0+2 = 2$
And
$a_5 = a_4 + d$
$a_5 = 2+2 = 4$
Therefore, missing terms are   -2,0,2,4
AP series is -4,-2,0,2,4,6

Given AP series is
$\small \fbox { },\: 38,\; \fbox { },\: \fbox { },\: \fbox { },\; -22$
Here, $a_2 = 38 , n = 6 \ and \ a_6 = -22$
Now,
$a_2=a_1+d$
$a_1=a =38-d \ \ \ \ \ \ \ \ \ -(i)$
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow a_6 =38-d+(6-1)d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (using \ (i))$
$\Rightarrow -22-38-=-d+5d$
$\Rightarrow d = -\frac{60}{4} = - 15$
Now,
$a_2= a_1+d$
$a_1 = 38-(-15) = 38+15 = 53$
And
$a_3=a_2+d$
$a_3=38-15 = 23$
And
$a_4 = a_3+d$
$a_4 = 23-15 = 8$
And
$a_5 = a_4 + d$
$a_5 =8-15 = -7$
Therefore, missing terms are   53,23,8,-7
AP series is 53,38,23,8,-7,-22

Given AP is
$\small 3,8,13,18,...,$
Let suppose that nth term of AP is 78
Here, $a = 3$
And
$d = a_2-a_1 = 8 - 3 = 5$
Now, we know that that
$a_n = a + (n-1)d$
$\Rightarrow 78 = 3 + (n-1)5$
$\Rightarrow 78 -3 = 5n-5$
$\Rightarrow n = \frac{75 +5}{5}= \frac{80}{5} = 16$
Therefore, value of  16th term of given AP is 78

Given AP series is
$\small 7,13,19,...,205$
Let's suppose there are n terms in given AP
Then,
$a = 7 , a_n = 205$
And
$d= a_2-a_1 = 13-7 = 6$
Now, we know that
$a_n =a + (n-1)d$
$\Rightarrow 205=7 + (n-1)6$
$\Rightarrow 205-7 = 6n-6$
$\Rightarrow n = \frac{198+6}{6} = \frac{204}{6} = 34$
Therefore, there are 34 terms in given AP

Given AP series is
$\small 18,15\frac{1}{2},13,...,-47$
suppose there are n terms in given AP
Then,
$a = 18 , a_n = -47$
And
$d= a_2-a_1 = \frac{31}{2}-18 = \frac{31-36}{2} = -\frac{5}{2}$
Now, we know that
$a_n =a + (n-1)d$
$\Rightarrow -47=18 + (n-1)\left ( -\frac{5}{2} \right )$
$\Rightarrow -47-18= -\frac{5n}{2}+\frac{5}{2}$
$\Rightarrow -\frac{5n}{2}= -65-\frac{5}{2}$
$\Rightarrow -\frac{5n}{2}= -\frac{135}{2}$
$\Rightarrow n = 27$
Therefore, there are 27 terms in given AP

Given AP series is
$\small 11,8,5,2...$
Here, $a = 11$
And
$d = a_2-a_1 = 8-11 = -3$
Now,
suppose -150 is nth term of the given AP
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow -150 = 11+(n-1)(-3)$
$\Rightarrow -150- 11=-3n+3$
$\Rightarrow =n = \frac{161+3}{3}= \frac{164}{3} = 54.66$
Value of n is not an integer
Therefore, -150 is not a term of AP $\small 11,8,5,2...$

It is given that
$\small 11$th term of an AP is  $\small 38$ and the $\small 16$th term is $\small 73$
Now,
$a_{11} =38= a+ 10d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
And
$a_{16} =73= a+ 15d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)$
On solving equation (i) and (ii) we will get
$a= -32 \ \ \ and \ \ \ d = 7$
Now,
$a_{31} = a+30d = -32 + 30\times 7 = -32+210 = 178$
Therefore, 31st terms of given AP is 178

It is given that
AP consists of $\small 50$ terms of  which $\small 3$rd term is $\small 12$ and the last term is $\small 106$
Now,
$a_3 = 12=a+2d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
And
$a_{50} = 106=a+49d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)$
On solving equation (i) and (ii) we will get
$a= 8 \ \ \ and \ \ \ d = 2$
Now,
$a_{29} = a+28d=8+28\times 2 = 8 +56 = 64$
Therefore, 29th term of given AP is 64

It is given that
$\small 3$rd and the $\small 9$th terms of an AP are $\small 4$  and  $\small -8$  respectively
Now,
$a_3 = 4=a+2d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
And
$a_{9} = -8=a+8d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)$
On solving equation (i) and (ii) we will get
$a= 8 \ \ \ and \ \ \ d = -2$
Now,
Let nth term of given AP is 0
Then,
$a_{n} = a+(n-1)d$
$0 = 8+(n-1)(-2)$
$2n = 8+2= 10$
$n = \frac{10}{2} = 5$
Therefore, 5th term of given AP is 0

It is given that
$\small 17$th term of an AP exceeds its $\small 10$th term by $\small 7$
i.e.
$a_{17}= a_{10}+7$
$\Rightarrow a+16d = a+9d+7$
$\Rightarrow a+16d - a-9d=7$
$\Rightarrow 7d=7$
$\Rightarrow d = 1$
Therefore, the common difference of AP is 1

Given  AP is
$\small 3,15,27,39,...$
Here, $a= 3$
And
$d= a_2-a_1 = 15 - 3 = 12$
Now, let's suppose nth term of given AP is  $\small 132$ more than its  $\small 54$th  term
Then,
$a_n= a_{54}+132$
$\Rightarrow a+(n-1)d = a+53d+132$
$\Rightarrow 3+(n-1)12 = 3+53\times 12+132$
$\Rightarrow 12n = 3+636+132+12$
$\Rightarrow 12n = 636+132+12$
$\Rightarrow n = \frac{780}{12}= 65$
Therefore, 65th term of given AP is  $\small 132$ more than its  $\small 54$th  term

It is given that
Two APs have the same common difference and  difference between their   $\small 100$th terms is  $\small 100$
i.e.
$a_{100}-a'_{100}= 100$
Let common difference of both the AP's is d
$\Rightarrow a+99d-a'-99d=100$
$\Rightarrow a-a'=100 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - (i)$
Now, difference between 1000th term is
$a_{1000}-a'_{1000}$
$\Rightarrow a+999d -a'-999d$
$\Rightarrow a-a'$
$\Rightarrow 100 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (using \ (i) )$
Therefore, difference between 1000th term is 100

We know that the first three digit number divisible by 7 is 105 and last three-digit number divisible by 7 is 994
Therefore,
$a = 105 , d = 7 \ and \ a_n = 994$
Let there are n three digit numbers divisible by 7
Now, we know that
$a_n = a+ (n-1)d$
$\Rightarrow 994 = 105 + (n-1)7$
$\Rightarrow 7n = 896$
$\Rightarrow n = \frac{896}{7} = 128$
Therefore,  there are 128 three-digit numbers divisible by 7

We know that the first number divisible by 4 between 10 to 250 is 12 and last  number divisible by 4 is 248
Therefore,
$a = 12 , d = 4 \ and \ a_n = 248$
Let there are n  numbers divisible by 4
Now, we know that
$a_n = a+ (n-1)d$
$\Rightarrow 248 = 12 + (n-1)4$
$\Rightarrow 4n = 240$
$\Rightarrow n = \frac{240}{4} = 60$
Therefore,  there are 60  numbers between 10 to 250 that are divisible by 4

Given two AP's are
$\small 63,65,67,...$   and    $\small 3,10,17,...$
Let first term and the common difference of two AP's are a , a' and d , d'
$a = 63 \ , d = a_2-a_1 = 65-63 = 2$
And
$a' = 3 \ , d' = a'_2-a'_1 = 10-3 = 7$
Now,
Let nth term of  both the AP's are equal
$a_n = a'_n$
$\Rightarrow a+(n-1)d=a'+(n-1)d'$
$\Rightarrow 63+(n-1)2=3+(n-1)7$
$\Rightarrow 5n=65$
$\Rightarrow n=\frac{65}{5} = 13$
Therefore, the 13th term of both the AP's are equal

It is given that
3rd term of AP is $\small 16$ and the $\small 7$th term exceeds the $\small 5$th term by $\small 12$
i.e.
$a_3=a+2d = 16 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
And
$a_7=a_5+12$
$a+6d=a+4d+12$
$2d = 12$
$d = 6$
Put the value of d in equation (i) we will get
$a = 4$
Now, AP with first term = 4 and common difference = 6 is
4,10,16,22,.....

Given AP is
$\small 3,8,13,...,253$
Here, $a = 3 \ and \ a_n = 253$
And
$d = a_2-a_1=8-3 = 5$
Let suppose there are n terms in the AP
Now, we know that
$a_n = a+(n-1)d$
$253= 3+(n-1)5$
$5n = 255$
$n = 51$
So, there are 51 terms in the given AP and 20th term from the last  will be 32th term from the starting
Therefore,
$a_{32} = a+31d$
$a_{32} = 3+31\times 5 = 3+155 = 158$
Therefore, 20th term from the of given AP is 158

It is given that
sum of the  $\small 4$th and  $\small 8$th terms of an AP is  $\small 24$ and the sum of the $\small 6$th and $\small 10$th terms is $\small 44$
i.e.
$a_4+a_8=24$
$\Rightarrow a+3d+a+7d=24$
$\Rightarrow 2a+10d=24$
$\Rightarrow a+5d=12 \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
And
$a_6+a_{10}=44$
$\Rightarrow a+5d+a+9d=44$
$\Rightarrow 2a+14d=44$
$\Rightarrow a+7d=22 \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)$
On solving equation (i) and (ii) we will get
$a= -13 \ and \ d= 5$
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, $a = 5000 \ and \ d =200$
Let's suppose after n years his salary will be Rs 7000
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow 7000=5000+(n-1)200$
$\Rightarrow 2000=200n-200$
$\Rightarrow 200n=2200$
$\Rightarrow n = 11$
Therefore, after 11years  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 $\small 1.75$
Therefore, $a = 5 \ and \ d = 1.75$
after  $\small n$th week, her weekly savings become Rs $\small 20.75$
Now, we know that
$a_n = a +(n-1)d$
$\Rightarrow 20.75= 5+(n-1)1.75$
$\Rightarrow 15.75= 1.75n-1.75$
$\Rightarrow 1.75n=17.5$
$\Rightarrow n=10$
Therefore, after 10 weeks her saving will become Rs 20.75

## NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions Excercise: 5.3

Given AP is
$\small 2,7,12,...,$ to $\small 10$ terms
Here, $a = 2 \ and \ n = 10$
And
$d = a_2-a_1=7-2=5$
Now, we know that
$S = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S = \frac{10}{2}\left \{ 2\times 2 +(10-1)5\right \}$
$\Rightarrow S = 5\left \{ 4 +45\right \}$
$\Rightarrow S = 5\left \{ 49\right \}$
$\Rightarrow S =245$
Therefore, the sum of AP  $\small 2,7,12,...,$ to $\small 10$ terms is 245

Given AP is
$\small -37,-33,-29,...,$ to $\small 12$ terms.
Here, $a = -37 \ and \ n = 12$
And
$d = a_2-a_1=-33-(-37)=4$
Now, we know that
$S = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S = \frac{12}{2}\left \{ 2\times (-37) +(12-1)4\right \}$
$\Rightarrow S = 6\left \{ -74 +44\right \}$
$\Rightarrow S = 5\left \{ -30\right \}$
$\Rightarrow S =-180$
Therefore, the sum of AP  $\small -37,-33,-29,...,$ to $\small 12$ terms. is -180

Given AP is
$\small 0.6,1.7,2.8,...,$ to $\small 100$  terms..
Here, $a = 0.6 \ and \ n = 100$
And
$d = a_2-a_1=1.7-0.6=1.1$
Now, we know that
$S = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S = \frac{100}{2}\left \{ 2\times (0.6) +(100-1)(1.1)\right \}$
$\Rightarrow S = 50\left \{ 1.2 +108.9\right \}$
$\Rightarrow S = 50\left \{ 110.1\right \}$
$\Rightarrow S =5505$
Therefore, the sum of AP $\small 0.6,1.7,2.8,...,$ to $\small 100$  terms. is 5505

Given AP is
$\small \frac{1}{15},\frac{1}{12},\frac{1}{10},...,$  to   $\small 11$  terms.
Here, $a = \frac{1}{15} \ and \ n = 11$
And
$d = a_2-a_1=\frac{1}{12}-\frac{1}{15}= \frac{5-4}{60}= \frac{1}{60}$
Now, we know that
$S = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S = \frac{11}{2}\left \{ 2\times \frac{1}{15} +(11-1)(\frac{1}{60})\right \}$
$\Rightarrow S = \frac{11}{2}\left \{ \frac{2}{15} +\frac{1}{6}\right \}$
$\Rightarrow S = \frac{11}{2}\left \{ \frac{9}{30}\right \}$
$\Rightarrow S =\frac{99}{60}= \frac{33}{20}$
Therefore, the sum of AP  $\small \frac{1}{15},\frac{1}{12},\frac{1}{10},...,$  to   $\small 11$  terms. is $\frac{33}{20}$

Given AP is
$\small 7+10\frac{1}{2}+14+...+84$
We first need to find the number of terms
Here, $a = 7 \ and \ a_n = 84$
And
$d = a_2-a_1=\frac{21}{2}-7= \frac{21-14}{2}= \frac{7}{2}$
Let suppose there are n terms in the AP
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow 84 = 7 + (n-1)\frac{7}{2}$
$\Rightarrow \frac{7n}{2}= 77+\frac{7}{2}$
$\Rightarrow n = 23$
Now, we know that
$S = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S = \frac{23}{2}\left \{ 2\times7 +(23-1)(\frac{7}{2})\right \}$
$\Rightarrow S = \frac{23}{2}\left \{ 14 +77\right \}$
$\Rightarrow S = \frac{23}{2}\left \{ 91\right \}$
$\Rightarrow S =\frac{2093}{2}=1046\frac{1}{2}$
Therefore, the sum of AP $\small 7+10\frac{1}{2}+14+...+84$   is $1046\frac{1}{2}$

Given AP is
$\small 34+32+30+...+10$
We first need to find the number of terms
Here, $a = 34 \ and \ a_n = 10$
And
$d = a_2-a_1=32-34=-2$
Let suppose there are n terms in the AP
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow 10 = 34 + (n-1)(-2)$
$\Rightarrow -26 = -2n$
$\Rightarrow n = 13$
Now, we know that
$S = \frac{n}{2}\left \{ a+a_n \right \}$

$\Rightarrow S = \frac{13}{2}\left \{ 44\right \}$
$\Rightarrow S =13\times 22 = 286$
Therefore, the sum of AP $\small 34+32+30+...+10$   is 286

Given AP is
$\small -5+(-8)+(-11)+...+(-230)$
We first need to find the number of terms
Here, $a = -5 \ and \ a_n = -230$
And
$d = a_2-a_1=-8-(-5)= -3$
Let suppose there are n terms in the AP
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow -230 = -5 + (n-1)(-3)$
$\Rightarrow -228 = -3n$
$\Rightarrow n = 76$
Now, we know that
$S = \frac{n}{2}\left \{ a+a_n \right \}$
$\Rightarrow S = \frac{76}{2}\left \{ (-5-230 )\right \}$

$\Rightarrow S = 38\left \{ -235\right \}$
$\Rightarrow S = -8930$
Therefore, the sum of AP $\small -5+(-8)+(-11)+...+(-230)$   is -8930

It is given that
$a = 5, d = 3 \ and \ a_n = 50$
Let suppose there are n terms in the AP
Now, we know that
$a_n = a+(n-1)d$
$\Rightarrow 50 = 5 + (n-1)3$
$\Rightarrow 48 = 3n$
$\Rightarrow n = 16$
Now, we know that
$S = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S = \frac{16}{2}\left \{ 2\times(5) +(16-1)(3)\right \}$
$\Rightarrow S = 8\left \{ 10+45\right \}$
$\Rightarrow S = 8\left \{ 55\right \}$
$\Rightarrow S =440$
Therefore, the sum of the given AP  is 440

It is given that
$a = 7 \ and \ a_{13} = 35$
$a_{13}= a+12d = 35$
$= 12d = 35-7 = 28$
$d = \frac{28}{12}= \frac{7}{3}$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{13} = \frac{13}{2}\left \{ 2\times(7) +(13-1)(\frac{7}{3})\right \}$
$\Rightarrow S_{13} = \frac{13}{2}\left \{14 +28\right \}$
$\Rightarrow S_{13} = \frac{13}{2}\left \{42\right \}$
$\Rightarrow S_{13} = 13 \times 21 = 273$
Therefore, the sum of given AP  is 273

It is given that
$d = 3 \ and \ a_{12} = 37$
$a_{12}= a+11d = 37$
$= a= 37-11\times 3 = 37-33=4$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{12} = \frac{12}{2}\left \{ 2\times(4) +(12-1)3\right \}$
$\Rightarrow S_{12} = 6\left \{ 8+33\right \}$
$\Rightarrow S_{12} = 6\left \{41\right \}$
$\Rightarrow S_{12} =246$
Therefore, the sum of given AP  is 246

It is given that
$\small a_3=15, S_1_0=125$
$a_{3}= a+2d = 15 \ \ \ \ \ \ \ \ -(i)$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{10} = \frac{10}{2}\left \{ 2\times(a) +(10-1)d\right \}$
$\Rightarrow 125 = 5\left \{ 2a+9d\right \}$
$\Rightarrow 2a+9d = 25 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)$
On solving equation (i) and (ii) we will get
$a= 17 \ and \ d = -1$
Now,
$a_{10} = a+ 9d = 17 + 9(-1)= 17-9 = 8$
Therefore, the value of d and 10th terms is -1 and 8 respectively

It is given that
$\small d=5, S_9=75$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{9} = \frac{9}{2}\left \{ 2\times(a) +(9-1)5\right \}$
$\Rightarrow 75= \frac{9}{2}\left \{ 2a +40\right \}$
$\Rightarrow 150= 18a+360$
$\Rightarrow a = -\frac{210}{18}=-\frac{35}{3}$
Now,
$a_{9} = a+ 8d = -\frac{35}{3} + 8(5)= -\frac{35}{3}+40 = \frac{-35+120}{3}= \frac{85}{3}$

It is given that
$\small a=2,d=8,S_n=90,$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow 90 = \frac{n}{2}\left \{ 2\times(2) +(n-1)8\right \}$
$\Rightarrow 180 = n\left \{ 4+8n-8\right \}$
$\Rightarrow 8n^2-4n-180=0$
$\Rightarrow 4(2n^2-n-45)=0$
$\Rightarrow 2n^2-n-45=0$
$\Rightarrow 2n^2-10n+9n-45=0$
$\Rightarrow (n-5)(2n+9)=0$
$\Rightarrow n = 5 \ \ and \ \ n = - \frac{9}{2}$
n can not be negative so the only the value of n is 5
Now,
$a_{5} = a+ 4d = 2+4\times 8 = 2+32 = 34$
Therefore, value of n and nth term is 5 and 34 respectively

It is given that
$\small a=8,a_n=62,S_n=210,$
Now, we know that
$a_n = a+(n-1)d$
$62 = 8+(n-1)d$
$(n-1)d= 54 \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$

Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow 210 = \frac{n}{2}\left \{ 2\times(8) +(n-1)d\right \}$
$\Rightarrow 420 = n\left \{ 16+54 \right \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (using \ (i))$
$\Rightarrow 420 = n\left \{ 70 \right \}$
$\Rightarrow n = 6$
Now, put this value in (i) we will get
$d = \frac{54}{5}$
Therefore, value of n and d are $6 \ and \ \frac{54}{5}$ respectively

It is given that
$\small a_n=4,d=2,S_n=-14,$
Now, we know that
$a_n = a+(n-1)d$
$4 = a+(n-1)2$
$a+2n = 6\Rightarrow a = 6-2n \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$

Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow -14 = \frac{n}{2}\left \{ 2\times(a) +(n-1)2\right \}$
$\Rightarrow -28 = n\left \{ 2(6-2n)+2n-2 \right \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (using \ (i))$
$\Rightarrow -28 = n\left \{ 10-2n \right \}$
$\Rightarrow 2n^2-10n-28=0$
$\Rightarrow 2(n^2-5n-14)=0$
$\Rightarrow n^2-7n+2n-14=0$
$\Rightarrow(n+2)(n-7)=0$
$\Rightarrow n = -2 \ \ and \ \ n = 7$
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

It is given that
$\small a=3,n=8,S=192,$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow 192 = \frac{8}{2}\left \{ 2\times(3) +(8-1)d\right \}$
$\Rightarrow 192=4\left \{6 +7d\right \}$
$\Rightarrow 7d = 48-6$
$\Rightarrow d = \frac{42}{7} = 6$
Therefore, the value of d is 6

It is given that
$\small l=28,S=144 \ and \ n = 9$
Now, we know that
$l = a_n = a+(n-1)d$
$28 = a_n = a+(n-1)d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow 144 = \frac{9}{2}\left \{ a + a +(n-1)d\right \}$
$\Rightarrow 288 =9\left \{ a+ 28\right \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(using \ (i))$
$\Rightarrow a+28= 32$
$\Rightarrow a=4$
Therefore, the value of a is 4

Given AP is
$\small 9,17,25,...$
Here, $a =9 \ and \ d = 8$
And  $S_n = 636$
Now , we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow 636 = \frac{n}{2}\left \{ 18+(n-1)8 \right \}$
$\Rightarrow 1272 = n\left \{ 10+8n \right \}$
$\Rightarrow 8n^2+10n-1272=0$
$\Rightarrow 2(4n^2+5n-636)=0$
$\Rightarrow 4n^2+53n-48n-636=0$
$\Rightarrow (n-12)(4n+53)=0$
$\Rightarrow n = 12 \ \ and \ \ n = - \frac{53}{4}$
Value of  n  can not be negative so the only the value of n is 12
Therefore, the sum of 12 terms of AP  $\small 9,17,25,...$ must be taken to give a sum of $\small 636$.

It is given that
$\small a=5,a_n=45,S_n=400,$
Now, we know that
$a_n = a+(n-1)d$
$45 = 5+(n-1)d$
$(n-1)d= 40 \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$

Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow 400 = \frac{n}{2}\left \{ 2\times(5) +(n-1)d\right \}$
$\Rightarrow 800 = n\left \{ 10+40 \right \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (using \ (i))$
$\Rightarrow 800 = n\left \{ 50 \right \}$
$\Rightarrow n = 16$
Now, put this value in (i) we will get
$d = \frac{40}{15}= \frac{8}{3}$
Therefore, value of n and d are $16 \ and \ \frac{8}{3}$ respectively

It is given that
$\small a=17,l=350,d=9,$
Now, we know that
$a_n = a+(n-1)d$
$350 = 17+(n-1)9$
$(n-1)9 = 333$
$(n-1)=37$
$n = 38$

Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{38}= \frac{38}{2}\left \{ 2\times(17) +(38-1)9\right \}$
$\Rightarrow S_{38}= 19\left \{ 34 +333\right \}$
$\Rightarrow S_{38}= 19\left \{367\right \}$
$\Rightarrow S_{38}= 6973$
Therefore, there are 38 terms  and their sun is 6973

It is given that
$\small a_{22}=149,d=7,n = 22$
Now, we know that
$a_{22} = a+21d$
$149 = a+21\times 7$
$a = 149 - 147 = 2$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{22}= \frac{22}{2}\left \{ 2\times(2) +(22-1)7\right \}$
$\Rightarrow S_{22}= 11\left \{ 4 +147\right \}$
$\Rightarrow S_{22}= 11\left \{ 151\right \}$
$\Rightarrow S_{22}= 1661$
Therefore, there are 22 terms  and their sum is 1661

It is given that
$\small a_{2}=14,a_3=18,n = 51$
And $d= a_3-a_2= 18-14=4$
Now,
$a_2 = a+d$
$a= 14-4 = 10$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{51}= \frac{51}{2}\left \{ 2\times(10) +(51-1)4\right \}$
$\Rightarrow S_{51}= \frac{51}{2}\left \{ 20 +200\right \}$
$\Rightarrow S_{51}= \frac{51}{2}\left \{ 220\right \}$
$\Rightarrow S_{51}= 51 \times 110$
$\Rightarrow S_{51}=5304$
Therefore, there are 51 terms  and their sum is 5610

It is given that
$S_7 = 49 \ and \ S_{17}= 289$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{7}= \frac{7}{2}\left \{ 2\times(a) +(7-1)d\right \}$
$\Rightarrow 98= 7\left \{ 2a +6d\right \}$
$\Rightarrow a +3d = 7 \ \ \ \ \ \ \ -(i)$
Similarly,
$\Rightarrow S_{17}= \frac{17}{2}\left \{ 2\times(a) +(17-1)d\right \}$
$\Rightarrow 578= 17\left \{ 2a +16d\right \}$
$\Rightarrow a +8d = 17 \ \ \ \ \ \ \ -(ii)$
On solving equation (i) and (ii) we will get
a = 1 and d = 2
Now, the sum of first n terms is
$S_n = \frac{n}{2}\left \{ 2\times 1 +(n-1)2 \right \}$
$S_n = \frac{n}{2}\left \{ 2 +2n-2 \right \}$
$S_n = n^2$
Therefore, the sum of n terms  is  $n^2$

It is given that
$\small a_n=3+4n$
We will check values of $a_n$ for different values of n
$a_1 = 3+4(1) =3+4= 7$
$a_2 = 3+4(2) =3+8= 11$
$a_3 = 3+4(3) =3+12= 15$
and so on.
From the above, we can clearly see that this is an AP with the first term(a) equals to and common difference (d) equals to 4
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 2\times(7) +(15-1)4\right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 14 +56\right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 70\right \}$
$\Rightarrow S_{15}= 15 \times 35$
$\Rightarrow S_{15}= 525$
Therefore, the sum of 15 terms  is  525

It is given that
$\small a_n=9-5n$
We will check values of $a_n$ for different values of n
$a_1 = 9-5(1) =9-5= 4$
$a_2 = 9-5(2) =9-10= -1$
$a_3 = 9-5(3) =9-15= -6$
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
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 2\times(4) +(15-1)(-5)\right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 8 -70\right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ -62\right \}$
$\Rightarrow S_{15}= 15 \times (-31)$
$\Rightarrow S_{15}= -465$
Therefore, the sum of 15 terms is -465

It is given that
the sum of the first $\small n$ terms of an AP is  $\small 4n-n^2$
Now,
$\Rightarrow S_n = 4n-n^2$
Now, first term is
$\Rightarrow S_1 = 4(1)-1^2=4-1=3$
Therefore, first term is 3
Similarly,
$\Rightarrow S_2 = 4(2)-2^2=8-4=4$
Therefore, sum of first two terms is 4
Now, we know that
$\Rightarrow S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_2 = \frac{2}{2}\left \{ 2\times 3+(2-1)d \right \}$
$\Rightarrow 4 = \left \{6+d \right \}$
$\Rightarrow d = -2$
Now,
$a_2= a+d = 3+(-2 )= 1$
Similarly,
$a_3= a+2d = 3+2(-2 )= 3-4=-1$
$a_{10}= a+9d = 3+9(-2 )= 3-18=-15$
$a_{n}= a+(n-1)d = 3+(n-1)(-2 )= 5-2n$

Positive integers divisible by 6 are
6,12,18,...
This is an AP with
$here, \ a = 6 \ and \ d = 6$
Now, we know that
$S_n= \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{40}= \frac{40}{2}\left \{ 2\times 6+(40-1)6 \right \}$
$\Rightarrow S_{40}= 20\left \{12+234 \right \}$
$\Rightarrow S_{40}= 20\left \{246 \right \}$
$\Rightarrow S_{40}= 4920$
Therefore,  sum of the first $\small 40$ positive integers divisible by  $\small 6$ is 4920

First 15 multiples of 8 are
8,16,24,...
This is an AP with
$here, \ a = 8 \ and \ d = 8$
Now, we know that
$S_n= \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 2\times 8+(15-1)8 \right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 16+112 \right \}$
$\Rightarrow S_{15}= \frac{15}{2}\left \{ 128 \right \}$
$\Rightarrow S_{15}= 15 \times 64 = 960$
Therefore,  sum of the first 15 multiple of 8 is 960

The odd number between 0 and 50 are
1,3,5,...49
This is an AP with
$here, \ a = 1 \ and \ d = 2$
There are total 25 odd number between 0 and 50
Now, we know that
$S_n= \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{25}= \frac{25}{2}\left \{ 2\times 1+(25-1)2 \right \}$
$\Rightarrow S_{25}= \frac{25}{2}\left \{ 2+48 \right \}$
$\Rightarrow S_{25}= \frac{25}{2}\times 50$
$\Rightarrow S_{25}= 25 \times 25 = 625$
Therefore,  sum of the odd numbers between $\small 0$ and $\small 50$  625

It is given that
Penalty for delay of completion beyond a certain date is Rs $\small 200$  for the first day, Rs $\small 250$ for the second day, Rs $\small 300$ for the third day and  penalty for each succeeding day being Rs $\small 50$ more than for the preceding day
We can clearly see that
200,250,300,..... is an AP  with
$a = 200 \ and \ d = 50$
Now, the penalty for 30 days is given by the expression
$S_{30}= \frac{30}{2}\left \{ 2\times 200+(30-1)50 \right \}$
$S_{30}= 15\left ( 400+1450 \right )$
$S_{30}= 15 \times 1850$
$S_{30}= 27750$
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 $S_7 = 700$
Let a be the prize money given to the 1st student
Then,
$S_7 = \frac{7}{2}\left \{ 2a+(7-1)(-20) \right \}$
$700 = \frac{7}{2}\left \{ 2a-120 \right \}$
$2a - 120 = 200$
$a = \frac{320}{2}= 160$
Therefore, the prize given to the first student is Rs 160
Now,
Let $a_2,a_2,...,a_7$ is the prize money given to the next 6 students
then,
$a_2 = a+d = 160+(-20)=160-20=140$
$a_3 = a+2d = 160+2(-20)=160-40=120$
$a_4 = a+3d = 160+3(-20)=160-60=100$
$a_5 = a+4d = 160+4(-20)=160-80=80$
$a_6 = a+5d = 160+5(-20)=160-100=60$
$a_7 = a+6d = 160+6(-20)=160-120=40$
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
$S_{12}= \frac{12}{2}\left \{ 2\times 3+(12-1)\times 3 \right \}$
$S_{12}= 6\left ( 6+33 \right )$
$S_{12}= 6 \times 39 = 234$
Therefore,  number of trees planted by 12 classes is 234

[Hint : Length of successive semicircles is  $\small l_1,l_2,l_3,l_4,...$  with centres at  $\small A,B,A,B,...,$ respectively.]

From the above-given figure

Circumference of 1st semicircle $l_1 = \pi r_1 = 0.5\pi$

Similarly,

Circumference of 2nd semicircle $l_2 = \pi r_2 = \pi$

Circumference of 3rd semicircle $l_3 = \pi r_3 = 1.5\pi$

It is clear that this is an AP with $a = 0.5\pi \ and \ d = 0.5\pi$

Now, sum of length of 13 such semicircles is given by

$S_{13} = \frac{13}{2}\left \{ 2\times 0.5\pi + (13-1)0.5\pi\right \}$
$S_{13} = \frac{13}{2}\left ( \pi+6\pi \right )$
$S_{13} = \frac{13}{2}\times 7\pi$
$S_{13} = \frac{91\pi}{2} = \frac{91}{2}\times \frac{22}{7}=143$
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  $a = 20 \ and \ d = -1$
Let suppose 200 logs are arranged in 'n' rows,
Then,
$S_n = \frac{n}{2}\left \{ 2\times 20 +(n-1)(-1) \right \}$
$200 = \frac{n}{2}\left \{ 41-n \right \}$
$\Rightarrow n^2-41n +400 = 0$
$\Rightarrow n^2-16n-25n +400 = 0$
$\Rightarrow (n-16)(n-25) = 0$
$\Rightarrow n = 16 \ \ and \ \ n = 25$
Now,
case (i) n = 25
$a_{25} =a+24d = 20+24\times (-1)= 20-24=-4$
But number of rows can not be in negative numbers
Therefore, we will reject the value n = 25

case (ii) n = 16

$a_{16} =a+15d = 20+15\times (-1)= 20-15=5$
Therefore, the number of rows in which 200 logs are arranged is equal to 5

A competitor starts from the bucket, picks up the nearest potato, runs back with it,  drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What  is the total distance the competitor has to run?

[Hint : To pick up the first potato and the second potato, the total distance (in metres) run by a competitor is  $\small 2\times5+2\times (5+3)$  ]

Distance traveled by the competitor in picking and dropping 1st potato $= 2 \times 5 = 10 \ m$

Distance traveled by the competitor in picking and dropping 2nd potato $= 2 \times (5+3) =2\times 8 = 16 \ m$

Distance traveled by the competitor in picking and dropping 3rd potato $= 2 \times (5+3+3) =2\times 11 = 22 \ m$

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 traveled by the competitor in picking and dropping  potatoes is
$S_{10}= \frac{10}{2}\left \{ 2\times 10+(10-1)6 \right \}$
$S_{10}= 5\left ( 20+54 \right )$
$S_{10}= 5\times 74 = 370$

Therefore, the total distance traveled by the competitor in picking and dropping  potatoes is  370 m

## NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions Excercise: 5.4

Given AP is
$\small 121,117,113,...,$
Here $a = 121 \ and \ d = -4$
Let suppose nth term of the AP is first negative term
Then,
$a_n = a+ (n-1)d$
If nth term is negative then $a_n < 0$
$\Rightarrow 121+(n-1)(-4) < 0$
$\Rightarrow 125<4n$
$\Rightarrow n > \frac{125}{4}=31.25$
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 $\small 8$
$a_3= a+ 2d$
$a_7= a+ 6d$
Now,
$a_3+a_7= a+ 2d+a+6d= 6$
$\Rightarrow 2a+8d = 6$
$\Rightarrow a+4d = 3 \Rightarrow a = 3-4d \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
And
$a_3.a_7 = (a+2d).(a+6d)=a^2+8ad +12d^2 = 8 \ \ \ \ \ \ \ -(ii)$
put value from equation (i) in (ii) we will get
$\Rightarrow (3-4d)^2+8(3-4d)d+12d^2= 8$
$\Rightarrow 9+16d^2-24d+24d-32d^2+12d^2=8$
$\Rightarrow 4d^2 = 1$
$\Rightarrow d = \pm \frac{1}{2}$
Now,
case (i)  $d = \frac{1}{2}$

$a= 3 - 4 \times \frac{1}{2} = 1$
Then,
$S_{16}=\frac{16}{2}\left \{ 2\times 1+(16-1)\frac{1}{2} \right \}$

$S_{16}=76$

case (ii)  $d = -\frac{1}{2}$
$a= 3 - 4 \times \left ( -\frac{1}{2} \right ) = 5$

Then,
$S_{16}=\frac{16}{2}\left \{ 2\times 1+(16-1)\left ( -\frac{1}{2} \right ) \right \}$

$S_{16}=20$

It is given that
The total distance between the top and bottom rung $= 2\frac{1}{2}\ m = 250cm$
Distance between any two rungs = 25 cm
Total number of rungs = $\frac{250}{25}+1= 11$
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 $a = 25 , a_{11} = 45 \ and \ n = 11$
Now, we know that
$a_{11}= a+ 10d$

$\Rightarrow 45=25+10d$
$\Rightarrow d = 2$
Now, total  length of the wood required for the rungs is equal to
$S_{11} = \frac{11}{2}\left \{ 2\times 25+(11-1)2 \right \}$
$S_{11} = \frac{11}{2}\left \{ 50+20 \right \}$
$S_{11} = \frac{11}{2}\times 70$
$S_{11} =385 \ cm$
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 $\small x$ 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
$S_n= \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
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.
$S_{n-1}=S_{49}-S_n$

$\frac{n-1}{2}\left \{ 2a+((n-1)-1)d \right \}=\frac{49}{2}\left \{ 2a+(49-1)d \right \}-\frac{n}{2}\left \{ 2a+(n-1)d \right \}$

$\frac{n-1}{2}\left \{ 2+(n-2) \right \}=\frac{49}{2}\left \{ 2+48 \right \}-\frac{n}{2}\left \{ 2+(n-1) \right \}$

$\frac{n-1}{2}\left \{ n\right \}=\frac{49}{2}\left \{ 50 \right \}-\frac{n}{2}\left \{ n+1 \right \}$

$\frac{n^2}{2}-\frac{n}{2}=1225-\frac{n^2}{2}-\frac{n}{2}$

$n^2 = 1225$
$n = \pm 35$

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  $\small 15$ steps each of which is $\small 50$ m long and Each step has a rise of   $\small \frac{1}{4}\: m$   and a tread of   $\small \frac{1}{2}\: m$
Now,
The volume required to make the first step = $\frac{1}{4}\times \frac{1}{2}\times 50 = 6.25 \ m^3$

Similarly,

The volume required to make 2nd step = $\left ( \frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{1}{2}\times \frac{1}{2}\times 50 = 12.5 \ m^3$
And
The volume required to make 3rd step = $\left ( \frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{3}{4}\times \frac{1}{2}\times 50 = 18.75 \ m^3$

And so on
We can clearly see that this is an AP with $a= 6.25 \ and \ d = 6.25$
Now, the total volume of concrete required to build the terrace of 15 such step is
$S_{15} =\frac{15}{2}\left \{ 2 \times 6.25 +(15-1)6.25 \right \}$
$S_{15} =\frac{15}{2}\left \{ 12.5 +87.5\right \}$
$S_{15} =\frac{15}{2}\times 100$
$S_{15} =15\times 50 = 750$
Therefore, the total volume of concrete required to build the terrace of 15 such steps is $750 \ m^3$

## NCERT solutions for class 10 maths chapter wise

 Chapter No. Chapter Name Chapter 1 CBSE NCERT solutions for class 10 maths chapter 1 Real Numbers Chapter 2 NCERT solutions for class 10 maths chapter 2 Polynomials Chapter 3 Solutions of NCERT class 10 maths chapter 3 Pair of Linear Equations in Two Variables Chapter 4 CBSE NCERT solutions for class 10 maths chapter 4 Quadratic Equations Chapter 5 NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions Chapter 6 Solutions of NCERT class 10 maths chapter 6 Triangles Chapter 7 CBSE NCERT solutions for class 10 maths chapter 7 Coordinate Geometry Chapter 8 NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry Chapter 9 Solutions of NCERT class 10 maths chapter 9 Some Applications of Trigonometry Chapter 10 CBSE NCERT solutions class 10 maths chapter 10 Circles Chapter 11 NCERT solutions  for class 10 maths chapter 11 Constructions Chapter 12 Solutions of NCERT class 10 chapter maths chapter 12 Areas Related to Circles Chapter 13 CBSE NCERT solutions class 10 maths chapter 13 Surface Areas and Volumes Chapter 14 NCERT solutions for class 10 maths chapter 14 Statistics Chapter 15 Solutions of NCERT class 10 maths chapter 15 Probability

## How to use NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions?

• First of all, 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.

Keep working hard & happy learning!