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 usingÂ Snâ€‹=2nâ€‹[2a+(nâˆ’1)d]Â we have,

S7â€‹=7â€‹/2[2a+(7âˆ’1)d]=49

â‡’49=7â€‹/2[2a+(7âˆ’1)d]

â‡’49=7/2â€‹(2a+6d)

â‡’7=a+3d

â‡’a+3d=7……………….(i)

S17â€‹=17/2â€‹[2a+(17âˆ’1)d]=289

â‡’289=17â€‹/2[2a+(17âˆ’1)d]

â‡’289=17/2â€‹(2a+16d)

â‡’17=a+8d

â‡’a+8d=17………………….(ii)

Substituting (i) from (ii), we get

5d=10Â orÂ d=2

From equation (i),

a+3(2)=7

a+6=7Â orÂ a=1

Snâ€‹=n/2â€‹[2(1)+(nâˆ’1)2]

=n/2â€‹[2+(nâˆ’1)2]

=n/2â€‹(2+2nâˆ’2)

=n^2