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$