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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 $\mathbf{d}=\mathbf{S}_{\mathbf{n}}-\mathbf{k} \mathbf{S}_{\mathbf{n}-\mathbf{1}}+\mathbf{S}_{\mathbf{n}-2}$ then k?

A. 1 B. 2 C. 3 D. None of these Answer: Option B
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Solution(By Apex Team)

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

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