Question

If the sum of first seven terms of an $A.P$ is $49$ and that of $17$ terms is $289.$ Find the sum of first $n$ terms .

Answer 1

Let first term of A.P is a

common difference = d

sum of n term in AP

$S=\frac{n}{2}[2a+(n-1\left)d\right]$ _________ (A)

$49=S_7=\frac{7}{2}[2a+6d]$ _______ (1) given

2a + 6d = 14

a + 3d = 7 _______ (2)

$S_{17}=\frac{17}{2}[2a+16d]=289$

$2(a+8d)=34$

a + 8d = 17 _________ (3)

equation (3) - equation (2)

5d = 10

d = 2

from equation (2)

a = 7 - 3d

a = 7 - 6 = 1

putting a & d values in equation (A)

$S_n=\frac{n}{2}[2\times 1+(n-1\left)2\right]$

$=n[1+n-1]=n^2$