Question

If $S_{1}$ is the sum of an arithmetic series of ' $n$ ' odd number of terms and $S_{2}$, the sum of the terms of the series in odd places, then $\frac{S_{1}}{S_{2}}=$

• A. $\frac{2 n}{n+1}$
• B. $\frac{n}{n+1}$
• C. $\frac{n+1}{2 n}$
• D. $\frac{n+1}{n}$

Answer 1

Answer: (A)

Let the odd number of terms of an arithmetic series be
$a, a+d, a+2 d, a+3 d, a+4 d, \ldots \ldots, a+(n-1) d$
Then,
$S_{1}=\frac{n}{2}\{2 a+(n-1) d\} $
$S_{2} =a+(a+2 d)+(a+4 d)+\ldots $ to $ \frac{n+1}{2}$ terms
$=\frac{n+1}{2 \times 2}\left[2 a+\left(\frac{n+1}{2}-1\right) \times 2 d\right]$
$=\frac{n+1}{4}(2 a+(n-1) d) $
$\therefore \frac{S_{1}}{S_{2}}=\frac{2 n}{n+1}$