Calculate $1+3+5+\ldots+2 n+1$.

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One can quickly conjecture that the result is $(n+1)^{2}$. We will prove this by induction.

The result is true for $n=0$, and if it is true for $n-1$, then:

$$
\begin{aligned}
1+3+5+\ldots+2 n+1 & =n^{2}+2 n+1 \\
& =(n+1)^{2}
\end{aligned}
$$

Hence the result.