It is given that sum of first 7 terms of an AP is equal to 49 and sum of first 17 terms is equal to 289.
Applying formula, Sn=n2(2a+ (n−1) d) to find sum of n terms of AP , we get
49=72(2a + (7−1) d)
⇒98=7(2a+6d)
⇒98=14a+42d
⇒7=a+3d
⇒a=7−3d (1)
And, 289=172(2a+ (17−1) d)
⇒578=17(2a+16d)
⇒34=2a+16d
⇒17=a+8d
Putting equation (1) in the above equation, we get
17=7−3d+8d
⇒10=5d
⇒d=105=2
Putting value of d in equation (1), we get
a=7−3d=7−3 (2) =7−6=1
Again applying formula, Sn=n2(2a+ (n−1) d) to find sum of n terms of AP , we get
Sn=n2[2(1) + (n−1) (2)]
⇒Sn=n2[2+2n−2]
⇒Sn=n2[2n]
⇒Sn= n2
Therefore, sum of n terms of AP is equal to n2