Question

If there are $(2n+1)$ terms in an arithmetic series, then prove that the ratio of the sum of odd terms to the sum of even terms is $(n+1):n$

Answer 1

We have to prove the ratio of sum of odd terms to sum of even terms of an arithmetic series to be $\frac{n+1}{n}$.

Since there are $2n+1$ terms there will be $n+1$ terms.

Let us consider the odd terms and even terms to be two different series.

These series will have common difference $2d$, where $d$ is the common difference of original series.

Let $a$ be the first term.

Sum of odd term series $=\frac{n+1}{2}(2a+n\times 2d)$ ....(i)

Sum of even term series $=\frac{n}{2}\left(2\right(a+d)+(n-1)\times 2d)=d\frac{n}{2}(2a+n\times 2d)$ ....(ii)

The ratio $\frac{\frac{n+1}{2}(2a+n\times 2d)}{\frac{n}{2}(2a+n\times 2d)}=\frac{n+1}{n}$