Question

If the sum of 1st 7 terms of an AP is 49 and that of 17 terms is 289 find the sum of 1st n terms​

Answer 1

Let a be the first term and d be the common difference .

We know that , $S_{n} = \frac{n}{2} [2a+(n-1)d]$

→  $S_{7} = \frac{7}{2} [2a+(7-1)d]$

→  $49 = \frac{7}{2} [2a+6d]$

→  $49 = 7(a+3d)$

→  $a+3d = 7$              ............... (1)

→  $S_{17} = \frac{17}{2} [2a+(17-1)d]$

→  $289 = \frac{17}{2} [2a+16d]$

→  $289=17(a+8d )$

→  $a+8d = 17$        ............... (2)

Eq(2) - Eq(1) ,

→ 5d = 10

→ d = 2

Putting the value of d in eq (1),

→ a +3×2 = 7

→ a = 1

$S_{n} = \frac{n}{2} [2\times1+(n-1)2]$

$S_{n} = n(1+n-1)$

$S_{n} = n\times n$

$S_{n} =n^{2}$

HOPE IT HELPS U ...............