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Math Help - One final geometric series question

  1. #1
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    One final geometric series question

    The sum to infinity of a geometric series with first term a and common ratio r is 10. The sum to infinity of a second geometric series with first term a and common ratio 2r is 15.
    a] find the value of r. [4]
    b] find the sum of the first four terms of the series, giving your answer correct to two decimal places. [3]

    Part b I will be okay with, just the first part is the trouble here.

    I got two simultaneous equations from it:

    10 - 10r = a - (1)
    15 - 30r = a - (2)

    Putting them together, I got 5 + 20r = a
    Or, 5 (1 + 4r) = a, whichever is preferred.

    So this means (if I'm right) that  \frac{a}{1 + 4r} = 5.

    If this is right, where do I go from here?
    If it is wrong, please could you help me?
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  2. #2
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    Quote Originally Posted by db5vry View Post
    The sum to infinity of a geometric series with first term a and common ratio r is 10. The sum to infinity of a second geometric series with first term a and common ratio 2r is 15.
    a] find the value of r. [4]
    I'm not sure where your equations came from...?

    Try using the infinite-summation formula:

    . . . . . \sum_{i=0}^{\infty}\,ar^i =\, \frac{a}{1\, -\, r}\, =\, 10

    . . . . . \sum_{i=0}^{\infty}\,a(2r)^i =\, \frac{a}{1\, -\, (2r)}\, =\, 15

    It is pretty easy to solve each of the above for "a=", so:

    . . . . . a\, =\, 10(1\, -\, r)\, =\, 15(1\, -\, 2r)

    Try solving the linear equation for the value of "r". You can then back-solve for the value of "a". :wink:
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