Question

If $S_1$ is the sum of an arithmetic progression 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. $\frac{2n}{n+1}$
• B. $\frac{n}{n+1}$
• C. $\frac{n+1}{2n}$
• D. $\frac{n+1}{n}$

Answer 1

Answer: (A)

$\frac{2n}{n+1}$

Let $n=2m+1$
Then $S_1=\frac{n}{2}[2a_1+(n-1)d]=\frac{2m+1}{2}[2a_1+2md]$
$=(2m+1)(a_1+md)=n(a+\frac{n-1}{2}d)$
$S_2=a_1+a_3+a_5+...+a_{2m+1}$
$=\frac{1}{2}\left(\frac{n+1}{2}\right)[2a_1+(\frac{n+1}{2}-1)2d]$
$=\left(\frac{n+1}{2}\right)[a_1+(\frac{n-1}{2}-1)d]$
$\therefore \frac{S_1}{S_2}=\frac{n}{\left(\frac{n+1}{2}\right)}=\frac{2n}{n+1}$