# Problem Of The Week #421 June 15th, 2020

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#### anemone

##### MHB POTW Director
Staff member
Here is this week's POTW:

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Solve for natural numbers for the identity below:

$\dfrac{1^4}{x}+\dfrac{2^4}{x+1}+\dfrac{3^4}{x+2}+\cdots+\dfrac{10^4}{x+9}=3025$

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#### anemone

##### MHB POTW Director
Staff member
Congratulations to the following members for their correct solution!

1. castor28
We use the fact that $\displaystyle\sum_{k=1}^n{k^3}=\frac{n^2(n+1)^2}{4}$.
For $x=1$, the LHS becomes:
$$1^3 + 2^3 + \cdots + 10^3 = \frac{10^2\cdot 11^2}{4}=3025$$
As this is the required value, $x=1$ is a solution. As the LHS is a decreasing function of $x$, this is the only solution.