Question

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):1

Answer 1

let us consider n=1 
then we get (2n+1)= 2*1+1=3
let us consider those 3 terms be 1,2,3 
sum of the odd terms 1+3=4
sum of the even terrms =2
Now the ratio of odd terms to even terms is 4:2=2:1
which satisfy the condition (n+1):1=(1+1):1=2:1
Hence proved

Answer 2

In AP. Let first term =a and common difference=d , Now, there are 2n+1 terms , i.e. no of terms is odd. Plz notice that in 2n+1 terms, n terms will be even while n+1 terms are odd
Now, consider the odd terms, Here, first term will be a, and common difference will be 2d
and for even numbered terms, First term will be a+d and common difference will be 2d

Now, coming to the problem, ,
Ratio of sum of odd terms to even terms=$\frac{ \frac{n+1}{2}(2a+(n+1-1)d }{ \frac{n}{2}(2(a+d)+(n-1)d) }$

Now, u can just simplify to see the desired ratio