Question:
Solution:
$\frac{1^{3}+2^{3}+3^{3}+\ldots+10^{3}}{1+2+\ldots+10}$
Since $1^{3}+2^{3}+\ldots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}$
$\Rightarrow 1^{3}+2^{3}+\ldots+10^{3}=\left[\frac{10(10+1)}{2}\right]^{2}$
$=\left[\frac{10 \times 11}{2}\right]^{2}$
$=(55)^{2}$
also $1+2+\ldots+n=\frac{n(n+1)}{2}$
$\Rightarrow 1+2+\ldots+10=\frac{10(11)}{2}=\frac{5 \times 11}{2}=55$
$\Rightarrow \frac{1^{3}+2^{3}+\ldots+10^{3}}{1+2+. .+n}=\frac{(55)^{2}}{55}=55$
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