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How to convert this loop into a closed form expression?

I have the following code in Python

    total=0
    for x in range(lowerBound,upperBound+1,2):
        total+=(x*x*x+5*x)/3+2

This iterates from x=lowerBound to the maximum value of x=upperBound (going by intervals of 2 at a time), tossing the value of $\lfloor(x^3+5x)/3\rfloor+2$ into the total (all division is floor division in Python unless using decimals, which I'd like to avoid if possible for precision's sake).

How can I convert this to a closed form expression?

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marked as duplicate by joriki, Martin Argerami, copper.hat, mixedmath Nov 26 '12 at 6:46

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1 Answer 1

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What you want is, for $L$ and $U$ gotten from your lower and upper bounds, $S = \sum_{x=L}^U \lfloor((2x)^3+5(2x))/3\rfloor+2$.

mod 3, $x^3+5x = x^3-x = x(x-1)(x+1)$, and one of these is always divisible by 3, so we can ignore the floor. You can also argue that $x^3+5x = 0, 6, $ and $18$ for $x = 0, 1, $ and $2$.

The $2$ contributes $2(U-L+1)$, so it will also be ignored.

We are left with $T = \sum_{x=L}^U ((2x)^3+5(2x))/3 = (1/3)\sum_{x=L}^U 8x^3+10x $.

Applying the usual formulae for the sums of integers and cubes will give you the answer.

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