Question

the sum of the first 5 terms of an ap and the sum of the first 7 terms of the same ap is 167 if the sum of the first 10 terms of this ap is 235 find the sum of the first 20 terms

Answer 1

Let the sum of first five terms be S5 and sum of the first seven terms be S7

We know that the formula for sum of n terms= n/2 ( 2a + (n-1)d )

So, S5 + S7= 167 (Given)
5/2 ( 2a + (5-1)d ) + 7/2 ( 2a + (7-1)d ) = 167
5/2 ( 2a + 4d ) + 7/2 (2a +6d) = 167
5a + 10d + 7a +21d = 167
12a + 31d = 167........... (1)

Now, It is also given that sum of first 10 terms is 235

So, 235= 10/2 (2a + 9d)
235= 5 ( 2a + 9d)
235 = 10 a + 45d
Dividing by 5
47= 2a + 9d............ (2)

Solving (1) and (2), we get

a=1 and d=5

So Sum of twenty terms will be

S20= 20/2 ( 2* 1 + (19)5 )
= 10 ( 2 + 95)
=970

Answer 2

answer is in attachment

hope it helps you