The sum of the first five terms of an AP and the sum of the first seven terms of theits first twenty terms.same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of first twenty terms

Explanation

Let the first term and common difference of A.P. be a and d respectively.
Now, `S_(5)+s_(7)=167`
`rArr (5)/(2)[2a+(5-1)d]+(7)/(2)[2a+(7-1)d]=167`
`rArr 5a+10d+7a+21d=167 rArr 12a+31d=167 " " ...(1)`
and `S_(10)=235`
`rArr (10)/(2)[2a+(10-1)d]=235`
`rArr 2a+9d=47 rArr 12a+54d=282 " " ...(2)`
Subtracting equation (1) from equation (2), we get
`{:(" "12a+54d=282),(underset(-)" "12a underset(-)+31d=""underset(-)(1)67),(bar(" "23d=115" ")rArrd=5):}`
Putting d=5 in equation (1), we get
`12a+31(5)=167`
`rArr 12a =12 rArr a=1`
Now `S_(20)=(20)/(2)[2a+(20-1)d]=10(2xx1+19xx5]=970`