Math

A series of 288 consecutive odd integers has a non-zero sum that is a perfect fourth power. Find the smallest possible sum for this series.

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  1. If the sum starts at the (k+1)st odd number, its sum is

    (k+288)^2 - k^2 = 576k + 82944

    When k=432, the sum is 331,776 = 24^4

    How you'd find that algebraically I have no clue.

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  2. aidan barkel om kapadia

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  3. a series of 288 consecutive odd integers has a non-zero sum that is a perfect fourth power, find the smallest possible sum for this series? (I hope I’m right idk how to explain this tho)
    a = smallest odd integer

    S = a + (a+2) + (a+4) + ... + (a+2*287) = 288(a+(a+574))/2 = 288(a+287)

    288 = 2^5 * 3^2

    Therefore, smallest value of (a+287) that will make S = 288(a+287) a perfect fourth power is:
    a + 287 = 2^3 * 3^2 = 72
    a = −215

    Smallest sum = 288*72 = 20736 = 12^4

    Smallest odd integer in series = −215
    −215 + −213 + −211 + ... + −1 + 1 + 3 + ... + 213 + 215 + ... + 359
    = 20736 = 12^4

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