Hi... Could anyone give me a logical explanation on how to change the order of summation as used in the problem: https://brilliant.org/discussions/thread/sequences-problem/ ?

Pls use examples of maybe any other series which uses the same concept..

Thanks in advance..

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## Comments

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TopNewestLet's say we want to add 1+4+9+16+25...225.

\(\sum^15_{r=1} r^2\)

Now we break the squares apart: 1+(1+3)+(1+3+5)+(1+3+5+7)+(1+3+...+9)...+(1+3+...29)

\(\sum^15_{r=1} \sum^r_{m=1} 2m-1\)

Now we rearrange everything: \(1 \times 15+3 \times 14+5 \times 13.....29 \times 1\)

\(\sum^15_{r=1} (2r-1)(16-r)\)

Well, I hope this answers that.

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can i

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