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

Since there are 2n+1 terms and there will be n+1 odd terms and n even 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 = $\frac{n+1}{2}(2 a+(n+1-1) \times 2 d)$

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

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

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