If $1^{3} + 2^{3} + 3^{3} + . . . . + n^{3} = 36100$ , then find $1 + 2 + 3 + . . . . + n$

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Solution

Given, $1^{3} + 2^{3} + 3^{3} + . . . . + n^{3} = 36100$
$\Rightarrow {(\frac{n (n + 1)}{2})}^{2} = 36100 = 19 \times 19 \times 10 \times 10$
$\Rightarrow \frac{n (n + 1)}{2} = 190$
Thus, $1 + 2 + 3 + . . . . + n = 190$

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