Question:
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
(a) 1
(b) 2
(c) 3
(d) none of these
Solution:
(b) 2
Let the A.P. be a, a+d, a+2d, a+3d...
Given:
$d=S_{n}-k S_{n-1}+S_{n-2}$
For n = 3, we have:
$d=(3 a+3 d)-k(2 a+d)+a$
$\Rightarrow 4 a+2 d-k(2 a+d)=0$
$\Rightarrow 2(2 a+d)=k(2 a+d)$
$\Rightarrow 2=k$
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