TeX source:

\begin{align*} 1+3+...+(2(n+1)-1) &= (1+3+...+(2n-1)) + (2(n+1)-1) \\ &= n^2 + (2n+1) \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \mathrm{by \; P(n) \; and \; basic \; algebra} \\ &= (n+1)^2 \blacksquare \end{align*}