Hello everyone, how are you today, hope you had an amazing Halloween!
I have this question i dont understand , i have looked up many websites but i dont understand their answers.
i would be grateful if any of you would explain it to me in detail.
If there are (2n+1) terms in an arithmetic series, prove that the ratio of the sum of odd place terms to the sum of even place terms is (n+1) : n .
Mmmmm....I'll try rosala ....!!!
Let's suppose that we have a partial series like this where a = the first term and d is the common difference between terms
a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) + (a + 5d) + (a + 6d) + (a + 7d) + (a + 8d)
Let n be the number of even place terms
And n + 1 the number of odd place terms
Then the sum of the even place terms is :
n *a + n^2* d = n (a + n*d)
And the sum of the odd place terms is :
(n + 1) * a + ( n+1) (n)* d = (n + 1) (a + n*d) for n ≥ 0
So....the ratio of the sum of the odd place terms to the even place terms is :
(n + 1) (a + n*d) ( n + 1)
_____________ = _______
n (a + n*d) n