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Show an identity using known identities


Solution: Recall from Exercise 1.4 that
\begin{equation}\label{eq:1-10-1}
1+3+\cdots+(2n-1)=n^2.
\end{equation} Note that
\begin{align*}
& (2n+1)+(2n+3)+\cdots+(4n-1)\\=&\ (1+3+5+\cdots+(4n-3)+(4n-1)) -(1+3+\cdots+(2n-1)).
\end{align*} Using \eqref{eq:1-10-1}, we find that
$$
1+3+5+\cdots+(4n-3)+(4n-1)=(2n)^2=4n^2,
$$ and
$$
1+3+\cdots+(2n-1)=n^2.
$$ Therefore
$$
(2n+1)+(2n+3)+\cdots+(4n-1)=4n^2-n^2=3n^2.
$$


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