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First, the idea of the argument. What happens when we increase \(n\) by 1? On the left-hand side, we increase the base of the square and go to the next square number. On the right-hand side, we increase the power of 2. This means we double the number. So the question is, how does doubling a number relate to increasing to the next square? Think about what the difference of two consecutive squares looks like. We have \((n+1)^2 - n^2\text{.}\) This factors:

\begin{equation*} (n+1)^2 - n^2 = (n+1-n)(n+1+n) = 2n+1\text{.} \end{equation*}

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