If $S_1$  is the sum of an arithmetic series of $n$  odd number of terms and $S_2$  is the sum of the terms of the series in odd places then $\frac{S_1}{S_2}=$

Let the odd number of terms of an arithmetic series be

$a,a+d,a+2d,a+3d,\mathrm{...........}a+\left(n-1\right)d$

$S_1=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

$S_2=a+a+2d+a+4d+\mathrm{...........}$  to $\frac{n+1}{2}$  terms

$=\frac{n+1}{2\times 2}\left[2a+\left(\frac{n+1}{2}-1\right)\times 2d\right]$

$=\frac{n+1}{4}\left(2a+\left(n-1\right)d\right)$

$\frac{S_1}{S_2}=\frac{2n}{n+1}$