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Math Help - Which Numbers in a Geometric Sequence Equal a Given Total

  1. #1
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    Which Numbers in a Geometric Sequence Equal a Given Total

    I have a sequence of positive integers. They were created using geometric progression: initial_value x (ratio^(nth-1))
    For example; if initial_value=1 , ratio=2 , and nth begins at 1 and increases +1 every iteration; I end up with a sequence of 1,2,4,8,16,32,64,...

    If, just for example, three integers from the above sequence were taken at random and added together, lets say they equaled 25(1+8+16).
    Given the sum, is there an equation that can give me the integers that were taken?
    For my particular situation, I will always have access to the initial_value, the ratio, and the nths used to create the sequence, the final maximum value of the sequence, and the sum of the integers taken.

    Thank you for your time.
    Last edited by AnOldMan; June 25th 2014 at 11:32 AM.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Which Numbers in a Geometric Sequence Equal a Given Total

    If the value of ratio is always an integer (is it?) then you can do this:

    The largest value that was used is A=Initial x R^[floor(log_{Base R}(N/initial))]. Here floor(x) means rounding x down to the nearest integer. Then subtract Initial x R^A and repeat. Keep going until the remainder is 0.

    Here's an example: suppose initial = 3, R= 2, and the sum is 33. The first value is 3 x 2^[floor(log_2 (33/3))] = 3 x 2^[floor(log_2 (11))] = 3 x 2^3 = 24. Now subtract 24 from 33, leaving 9. The next value is 3 x 2^[floor(log_2(9/3))] = 3 x 2^1 = 6. Subtract 6 from 9 leaving 3. The last value is 3 x 2^[floor(log_2(3/3)] = 3x2^0 = 3. Hence the three values used are 24, 6, and 3.

    Hope this helps.
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    Re: Which Numbers in a Geometric Sequence Equal a Given Total

    The value of ratio is always an integer, yes.

    Your method seems to work fine for me. Thank you very much for your time and help. I truly appreciate it.
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