Let S denotes 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 =

Answer: Option B

S_n is the sum of n terms of an A.P.
a is its first term and d is common difference
$$\begin{array}{rl}& d=S_n-k{S}_{n-1}+S_{n-2}\\ & \Rightarrow k{S}_{n-1}=S_n+S_{n-2}-d\\ & =\left(a_n+S_{n-1}\right)+\left(S_{n-1}-a_{n-1}-1\right)-d\end{array}$$

$$\left\{\begin{array}{l}\because S_n=S_{n-1}+a_n\mathrm{and}\\ S_{n-1}=a_{n-1}+S_{n-2}\\ \Rightarrow S_{n-2}=S_{n-1}-a_{n-1}\end{array}\right\}$$

$$\begin{array}{rl}& =a_n+2S_{n-1}-a_{n-1}-d\\ & =2S_{n-1}+a_n-a_{n-1}-d\\ & =2S_{n-1}+d-d\left(\because a_n-a_{n-1}=d\right)\\ & =2S_{n-1}\\ & \therefore k=2\end{array}$$