Question

Let $S_1,S_2,S_3$ be the sum of n terms of three series in A.P., the first term of each being 1 and the common differences 1, 2, 3 respectively. If $S_1+S_3=\lambda ,S_2$ , then the value of $\lambda$ is

• A. 1
• B. 2
• C. 3
• D. none of these.

Answer 1

Answer: (B)

$S_1=\frac{n}{2}\left[2+\left(n-1\right)1\right]=\frac{n}{2}\left(n+1\right)$
$S_2=\frac{n}{2}\left[2+\left(n-1\right)2\right]=n\left(n\right)=n^2$
$S_3=\frac{n}{2}\left[2+\left(n-1\right)3\right]=\frac{n}{2}\left[3n-1\right]$
Since $S_1+S_3=\lambda S_2$
$\therefore \frac{n}{2}\left[n+1+3n-1\right]=\lambda .n^2$
$\Rightarrow 2n^2=\lambda n^2$
$\Rightarrow \lambda =2$