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# help

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Hana writes the following sums:

1^3 = 1,

2^3 = 3 + 5,

3^3 = 7 + 9 + 11,

4^3 = 13 + 15 + 17 + 19

If the sum of n^3 contains the sum 2021, then what is n?

May 5, 2020

#1
+25451
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Hana writes the following sums:

$$1^3 = 1, \\ 2^3 = 3 + 5, \\ 3^3 = 7 + 9 + 11, \\ 4^3 = 13 + 15 + 17 + 19$$

If the sum of $$n^3$$ contains the sum 2021, then what is n?

I assume:

$$\begin{array}{|rcll|} \hline n^2-(n-1) ~\le~ 2021 ~\le~ n^2+(n-1) \\ \Rightarrow \mathbf{n=45} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 45^3 &=& 1981+1983+\ldots+{\color{red}2021}+\ldots+2025+\ldots+2068+2069 \\\\ && \boxed{2025=45^2 \\ 1981=45^2-(45-1) \\ 2069=45^2+(45-1) }\\ \hline \end{array}$$

May 5, 2020