Question

let S be the sum of the first n terms of an AP.S2n=3n+14n2 ,then what is the common diffrence?​

Answer 1

The common difference(d) is "7".

Step-by-step explanation:

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

Given,

$S_{2n}$ = 3n + 14$n^{2}$

Put n = 1, 2, 3, ....

∴ $S_{2 × 1}$ = 3 × 1 + 14$1^{2}$

⇒ $S_{2 × 1}$ = 3 × 1 + 14 × 1

⇒ $S_{2}$ = 17

⇒ $\dfrac{2}{2}$·{2a + (2 - 1)d} = 17

⇒ 2a + d =17   ....(1)

and

∴ $S_{2 × 2}$ = 3 × 2 + 14$2^{2}$

⇒ $S_{2 × 1}$ = 3 × 2 + 14 × 4 = 6 + 56 = 62

⇒ $S_{2}$ = 62

⇒ $\dfrac{4}{2}$·{2a + (4 - 1)d} = 62

⇒ 2·{2a + 3d} = 62

⇒ 2a + 3d = 31

⇒ (2a + d) + 2d = 31

⇒ 17 + 2d = 31 [Using equation (1)]

⇒  2d = 31  - 17 = 14

⇒  d = 7

Hence, the common difference(d) is "7".