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The common difference of an A.P., the sum of whose n terms is $\mathbf{s}_{\mathbf{n}}$, is

A. $\mathrm{S}_{\mathrm{n}}-2 \mathrm{~S}_{\mathrm{n}-1}+\mathrm{S}_{\mathrm{n}-2}$ B. $S_{n}-2 S_{n-1}-S_{n-2}$ C. $\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-2}$ D. $\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-1}$ Answer: Option A
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Solution(By Apex Team)

$\begin{array}{l} \text { Sum of } n \text { terms }=S_{n} \\ \therefore a_{n}=S_{n}-S_{n-1} \\ \text { and } a_{n-1}=S_{n-1}-S_{n-2} \\ \therefore \text { Common difference }(d)=a_{n}-a_{n-1} \\ =\left(S_{n}-S_{n-1}\right)-\left(S_{n-1}-S_{n-2}\right) \\ S_{n}-S_{n-1}-S_{n-1}+S_{n-2} \\ =S_{n}-2 S_{n-1}+S_{n-2} \end{array}$

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