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$11^{3}+12^{3}+13^{3}+\ldots . .+20^{3}$ is
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1239 Upvotes
Verified Answer
The correct answer is:
an odd integer divisible by 5 .
We have,
$$
\begin{aligned}
11^{3}+12^{3}+13^{3}+\ldots+20^{3} \\
=\left(1^{3}+2^{3}+\ldots+10^{3}+11^{3}+\right.&\left.12^{3}+\ldots .+20^{3}\right) \\
&-\left(1^{3}+2^{3}+\ldots .+10^{3}\right)
\end{aligned}
$$
$$
\begin{aligned}
&=\Sigma(20)^{3}-\Sigma(10)^{3} \\
&=\left(\frac{20(20+1)}{2}\right)^{2}-\left(\frac{10(10+1)}{2}\right)^{2} \\
&=(10 \times 21)^{2}-(5 \times 11)^{2}=(210-55)(210+55) \\
&=155 \times 265=41075=5 \times 8215
\end{aligned}
$$
$$
\begin{aligned}
11^{3}+12^{3}+13^{3}+\ldots+20^{3} \\
=\left(1^{3}+2^{3}+\ldots+10^{3}+11^{3}+\right.&\left.12^{3}+\ldots .+20^{3}\right) \\
&-\left(1^{3}+2^{3}+\ldots .+10^{3}\right)
\end{aligned}
$$
$$
\begin{aligned}
&=\Sigma(20)^{3}-\Sigma(10)^{3} \\
&=\left(\frac{20(20+1)}{2}\right)^{2}-\left(\frac{10(10+1)}{2}\right)^{2} \\
&=(10 \times 21)^{2}-(5 \times 11)^{2}=(210-55)(210+55) \\
&=155 \times 265=41075=5 \times 8215
\end{aligned}
$$
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