###
**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**

## Show Answer

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
$\begin{array}{l}d=S_n-kS_{n-1}+S_{n-2}&\\
\Rightarrow kS_{n-1}=S_n+S_{n-2}-d&\\
=\left(a_n+S_{n-1}\right)+\left(S_{n-1}-a_{n-1}-1\right)-d&\\
\quad\left\{\begin{array}{l}\because S_n=S_{n-1}+a_n\text{ and }\\
S_{n-1}=a_{n-1}+S_{n-2}\\
\Rightarrow S_{n-2}=S_{n-1}-a_{n-1}\end{array}\right\}&\\
&\\
=a_n+2S_{n-1}-a_{n-1}-d&\\
=2S_{n-1}+a_n-a_{n-1}-d&\\
=2S_{n-1}+d-d\left(\because a_n-a_{n-1}=d\right)&\\
=2S_{n-1}&\\
\therefore k=2&\end{array}$

## Related Questions On Progressions

### How many terms are there in 20, 25, 30 . . . . . . 140?

A. 22B. 25

C. 23

D. 24

### Find the first term of an AP whose 8th and 12th terms are respectively 39 and 59.

A. 5B. 6

C. 4

D. 3

### Find the 15th term of the sequence 20, 15, 10 . . .

A. -45B. -55

C. -50

D. 0

### The sum of the first 16 terms of an AP whose first term and third term are 5 and 15 respectively is

A. 600B. 765

C. 640

D. 680