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18.  The difference between any two consecutive interior angles of a polygon is $5 \degree$. If the smallest angle is $120 \degree$  , find the number of the sides of the polygon.

The angles of polygon forms AP with common difference of  and first term as  . We know that sum of angles of polygon with n sides is  Sides of polygon are 9 or 16.

17.   A man starts repaying a loan as first instalment of Rs. 100. If he increases the instalment by Rs 5 every month, what amount he will pay in the 30th instalment?

The  first instalment is of Rs. 100. If the instalment increase by Rs 5 every month, second instalment is Rs.105. Then , it forms an AP. We have , Thus, he will pay  Rs. 245 in the 30th instalment.

16.   Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A. P. and the ratio of $7 ^{th}$ and  $(m-1)^{th}$  numbers is 5 : 9. Find the  value of m.

Let A,B,C.........M be m numbers. Then,  Here we have,         Given : the ratio of and    numbers is 5 : 9. Putting value of d from above, Thus, value of m is 14.

15.   If   $\frac{a^n + b ^n }{a ^{ n-1}+ b ^{n-1}}$  is the A.M. between a and b, then find the value of n.

Given :    is the A.M. between a and b. Thus, value of n is 1.

14.   Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

Let five numbers be A,B,C,D,E. Then   Here we have,         Thus, we have                                                                                      Thus, the five numbers are 11,14,17,20,23.

13.  If the sum of n terms of an A.P. is  $3 n^2 + 5 n$  and its $m^{th }$ term is 164, find the value of m.

Given : If the sum of n terms of an A.P. is    and its term is 164 Let a and d be first term and common difference of a AP ,respectively.  Sum of n terms =  Comparing the coefficients of n on both side , we have       Also ,   m th term is 164. Hence, the value of m is 27.

12.   The ratio of the sums of m and n terms of an A.P. is $m^2 : n^2$ . Show that the ratio of mth and nth term is $( 2m-1) : ( 2n- 1 )$.

Let a and b be the first term and common difference of a AP ,respectively. Given : The ratio of the sums of m and n terms of an A.P. is  . To prove :  the ratio of mth and nth term is .   Put , we get  From equation (1) ,we get Hence proved.

11. Sum of the first p, q and r terms of an A.P. are a, b and c, respectively. Prove that

$\frac{a}{p} ( q-r ) + \frac{b}{q}( r-p ) + \frac{c}{r} ( p-q ) = 0$

To prove :  Let  and d be the first term and the common difference of AP, respectively. According to the given information, we have       Subtracting equation (2) from (1), we have  Subtracting equation (3) from (2), we have  Equating values of d, we have  Dividing both sides from pqr, we get  Hence, the given result is proved.

10.   If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

Let first term of AP = a and common difference = d. Then,  Given :    Now,                                                             Thus, sum of p+q terms of AP is 0.

9.  The sums of n terms of two arithmetic progressions are in the ratio $5n + 4 : 9n + 6$ . Find the ratio of their  18th terms.

Given: The sums of n terms of two arithmetic progressions are in the ratio. There are two AP's with first terms =    and common difference =   Substituting n=35,we get Thus, the ratio of the 18th term of AP's is

8.  If the sum of n terms of an A.P. is $( pn + qn ^ 2 )$ , where p and q are constants,  find the common difference

If the sum of n terms of an A.P. is , Comparing coefficients of  on both side , we get                  The common difference of AP is 2q.

7.  Find the sum to n terms of the A.P., whose $k^{th}$ term is 5k + 1.

Given :  Comparing LHS and RHS , we have         and      Putting value of d,

6. If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term.

Given : A.P. 25, 22, 19, …..... a=25  , d = -3 n could not be  so n=8. Last term                                                  The, last term of A.P. is 4.

5. In an A.P., if pth term is 1/q  and qth term is 1/p  , prove that the sum of first pq terms is 1/2  (pq +1), where $p \neq q$

Given : In an A.P., if pth term is 1/q  and qth term is 1/p  Subtracting (2) from (1), we get                Putting value of d in equation (1),we get  Hence,the sum of first pq  terms is 1/2  (pq +1), where .

4.  How many terms of the A.P. $-6 , -11/2 , -5...$  are needed to give the sum –25?

Given : A.P. =  Given : sum = -25

3.  In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.

First term =a=2 Let the series be  Sum of first five terms  Sum of next five terms  Given : The sum of the first five terms is one-fourth of  the next five terms. To prove  :  L.H.S :  Hence, 20th term is –112.

2.  Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

Numbers divisible by 5 from 100 to 1000 are  This sequence is an A.P. Here , first term =a =105 common difference = 5. We know ,                                                          The  sum of numbers divisible by 5 from 100 to 1000 is 98450.

1.  Find the sum of odd integers from 1 to 2001.

Odd integers from 1 to 2001 are  This sequence is an A.P. Here , first term =a =1 common difference = 2. We know ,                                                      The , sum of odd integers from 1 to 2001 is 1002001.
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