Thread: Need help with this geometric progression problem

1. Need help with this geometric progression problem

A geometric progression G has positive first term a, common ratio r (where 0< r < 1 ) and sum to infinity S. The sum to infinity of even-numbered terms of G, i.e. 2nd, 4th, 6th...terms is ((-1/2)S).
i)Find the value of r
ii) Given that the third term of G is 2, find that the sum to infinity of the odd-numbered terms of G i.e. 1st, 3rd, 5th... terms.
Thanks!

2. Originally Posted by azuki
A geometric progression G has positive first term a, common ratio r (where 0< r < 1 ) and sum to infinity S. The sum to infinity of even-numbered terms of G, i.e. 2nd, 4th, 6th...terms is ((-1/2)S).
i)Find the value of r
ii) Given that the third term of G is 2, find that the sum to infinity of the odd-numbered terms of G i.e. 1st, 3rd, 5th... terms.
Thanks!
(i) if the sum of the even terms is $-\frac{S}{2}$, then the sum of the odd terms is $\frac{3S}{2}$

$-\frac{S}{2} = ar + ar^3 + ar^5 + ... = r(a + ar^2 + ar^4 + ...) = r\left(\frac{3S}{2}\right)$

$r = -\frac{1}{3}$

you should now be able to answer (ii)

3. How do you know that the sum of the odd terms is ?

4. Originally Posted by azuki
How do you know that the sum of the odd terms is ?
Sum of even terms plus sum of odd terms = sum of all terms = S.