One of the most important and exam oriented question from arithmetic progression chapter in which we have been asked to find the sum of first n terms if the sum of first 7 terms of an AP is 49 and the sum of its first 17 terms is 289.

Book – RS Aggarwal, Class 10, chapter 5C, question no 29

By using Sn=2n[2a+(n−1)d] we have,

S7=7/2[2a+(7−1)d]=49

⇒49=7/2[2a+(7−1)d]

⇒49=7/2(2a+6d)

⇒7=a+3d

⇒a+3d=7……………….(i)

S17=17/2[2a+(17−1)d]=289

⇒289=17/2[2a+(17−1)d]

⇒289=17/2(2a+16d)

⇒17=a+8d

⇒a+8d=17………………….(ii)

Substituting (i) from (ii), we get

5d=10 or d=2

From equation (i),

a+3(2)=7

a+6=7 or a=1

Sn=n/2[2(1)+(n−1)2]

=n/2[2+(n−1)2]

=n/2(2+2n−2)

=n^2