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Problem Of The Week #421 June 15th, 2020

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
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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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
Congratulations to the following members for their correct solution! (Cool)

1. castor28
2. kaliprasad

Solution from 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.
 
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