### 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

### Solution(By Apex Team)

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

A. 22
B. 25
C. 23
D. 24

A. 5
B. 6
C. 4
D. 3

A. -45
B. -55
C. -50
D. 0