Let S_n denote the sum of n terms of an A.P. whose first term is …

MCQ

Let $S_n$ denote the sum of $n$ terms of an A.P. whose first term is a. If the common difference $d$ is given by $d=S_n-k$ $S_{n-1}+S_{n-2}$, then $k=$

A
1
✓
2
C
3
D
None of these

✓

Answer

Correct option: B.

Solution:
Let the A.P. be $\text{a},\ \text{a}+\text{d},\ \text{a}+\text{d},\ \text{a}+3\text{d}\ ...$
Given:
$\text{d}=\text{S}_\text{n}-\text{k}\text{S}_{\text{n}-1}+\text{S}_{\text{n}-2}$
For $\text{n}=3,$ we have
$\text{d}=(3\text{a}+3\text{d})-\text{k}(2\text{a}+\text{d})+\text{a}$
$\Rightarrow4\text{a}+2\text{d}=\text{k}(2\text{a}+\text{d})=0$
$\Rightarrow2(2\text{a}+\text{d})=\text{k}(2\text{a}+\text{d})$
$\Rightarrow2=\text{k}$

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