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Math Help - sum of squares of all numbers between 10 and 70 and which are divisible by 4.

  1. #1
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    sum of squares of all numbers between 10 and 70 and which are divisible by 4.

    Find the sum of squares of all numbers between 10 and 70 and which are divisible by 4.
    Last edited by mr fantastic; November 4th 2010 at 05:13 PM. Reason: Re-titled.
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  2. #2
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    All evens are divisible by 4; so "sum of even squares":

    n = 70/2 = 35; f = (10-2)/2 = 4

    2n(n+1)(2n+1)/3 - 2f(f+1)(2f+1)/3 = 59640 - 120 = 59520

    "See" that?
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  3. #3
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    Hello, rickrishav!

    I hope I interpreted the problem correctly . . .


    Find the sum of squares of all integers between 10 and 70 which are divisible by 4.

    The integers between 10 and 70 which are multiple of 4 are:

    . . \{12,16,20,24,\,\hdots\,68\} \;=\;\{3\!\cdot\!4,\:4\!\cdot\!4,\:5\!\cdot\!4,\:6  \!\cdot\!6,\:\hdots\:17\!\cdot\!4\}


    The sum of their squares is:

    . . S \;=\;3^2\!\cdot\!4^2 + 4^2\!\cdot\!4^2 + 5^2\!\cdot\!4^2 + 5^2\!\cdot4^2 + \hdots + 17^2\!\cdot\!4^2

    . . . . =\;4^2\left(3^2 + 4^2 + 5^2 + \hdots + 17^2\right)

    . . . . \displaystyle = \;4^2\left(\sum^{17}_{k=1} k^2 - \sum^2_{k=1}k^2\right)

    . . . . -\; 4^2\left(\dfrac{17\!\cdot\!18\!\cdot\!35}{6} - \dfrac{2\!\cdot\!3\!\cdot\!5}{6}\right) .**

    . . . . =\;16\cdot 1780

    . . . . =\;28,\!480


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    **

    The sum of the first \,n squares is:

    . . \displaystyle \sum^n_{k+1} k^2 \;=\; 1^2 + 2^2 + 3^2 + \hdots + n^2 \;=\;\frac{n(n+1)(2n+1)}{6}


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