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# Could anyone please tell me how do we change the order of summation?

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..

Note by Krishna Jha
4 years, 3 months ago

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Let'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. · 4 years, 3 months ago