Solve for natural numbers: (1^4 /z) + (2^4 /(z+1)) + (3^4 /(z+2)) + ... + ((10^4 )/(z+9)) = 3025


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Question Number 144553 by mathdanisur last updated on 26/Jun/21

$S o l v e f o r n a t u r a l n u m b e r s :$ $\frac{1^{4}}{z} + \frac{2^{4}}{z + 1} + \frac{3^{4}}{z + 2} + . . . + \frac{10^{4}}{z + 9} = 3025$
	Answered by Olaf_Thorendsen last updated on 26/Jun/21

	$S (z) = \underset{n = 0}{\sum^{9}} \frac{{(n + 1)}^{4}}{z + n}$ $S (1) = \underset{n = 0}{\sum^{9}} \frac{{(n + 1)}^{4}}{n + 1} = \underset{n = 0}{\sum^{9}} {(n + 1)}^{3} = \underset{n = 1}{\sum^{10}} n^{3}$ $S (1) = \frac{n^{2} {(n + 1)}^{2}}{4} ∣_{n = 10} = \frac{100 \times 121}{4} = 3025$ $\Rightarrow 1 is a solution .$ $if z ⩾ 2, S (z) ⩽ S (1) = 3025 : no solution$
	Commented by mathdanisur last updated on 26/Jun/21

	$c o o l t h a n k y o u S i r$
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