If-the-sum-of-n-terms-f-a-series-is-A-n-2-B-n-where-A-B-are-constants-then-it-is-an-AP-

Question Number 25741 by gugalesweta@gmail.com last updated on 13/Dec/17

If the sum of n terms f a series is

A n^{2} + B n, where A, B are constants,

then it is an AP .

Commented by math solver last updated on 13/Dec/17

y e s .

Answered by Rasheed.Sindhi last updated on 14/Dec/17

S_{n} = {An}^{2} + Bn

S_{n - 1} = A {(n - 1)}^{2} + B (n - 1)

T_{n} = ({An}^{2} + Bn) - {A {(n - 1)}^{2} + B (n - 1)}

= {An}^{2} - A {(n - 1)}^{2} + Bn - B (n - 1)

= {An}^{2} - A (n^{2} - 2 n + 1) + Bn - Bn + B

= 2 An - A + B

The difference between consecutive

terms

d = T_{k} - T_{k - 1} = 2 Ak - A + B - {2 A (k - 1) - A + B}

= 2 Ak - A + B - 2 A (k - 1) + A - B

= 2 Ak - 2 Ak + 2 A

= 2 A

Since the difference between any

two consecutive terms is constant

The given sequence is AP .

Commented by mrW1 last updated on 14/Dec/17

{T h a t}^{'} s g r e a t!

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