Question

Let $S_n$ denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by $d=S_n-k{S}_{n-1}+S_{n-2}$, then k =

• A. 1
• B. 2
• C. 3
• D. None of these

Answer 1

The correct option is B

2

Let the A.P. be $a,a+d,a+2d,a+3d\dots \dots$

Given:

$d=S_n-k{S}_{n-1}+S_{n-2}$

For n = 3, we have:

d = (3a + 3d) - k(2a + d) + a

$\Rightarrow 4a+2d-k(2a+d)+a$

$\Rightarrow 2(2a+d)=k(2a+d)=0$

$\Rightarrow 2=k$