Intuitive ways to get formula of cubic sum

We have $$ \sum_{k=1}^{n}k^3 = 1 + 8 + 27 + \ldots + n^3 = \\ \underbrace{1}_{1^3} + \underbrace{3+5}_{2^3} + \underbrace{7 + 9 + 11}_{3^3} + \underbrace{13 + 15 + 17 + 19}_{4^3} + \ldots = \\ \underbrace{\underbrace{\underbrace{1}_{1^2} + 3}_{2^2} + 5}_{3^2} + \ldots $$ which is $$ \left( \sum_{k=1}^{n}k \right)^2 $$


Maybe this will help you visualize it:

Source.

or this one which is clearer:

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Tags:

Sequences And Series

Summation

Algebra Precalculus

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