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Math Help - Need help with this geometric progression problem

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by azuki View Post
    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)
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  3. #3
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    How do you know that the sum of the odd terms is ?
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  4. #4
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    Quote Originally Posted by azuki View Post
    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.
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