\beginsection 1.3

Prove $1^3+2^3+\cdots+n^3=(1+2+\cdots+n)^2$ for all natural numbers $n$.

\medskip

Induction Step 1: Show that $P_1$ is true.
$$P_1=1^3=1^2$$
Induction Step 2: Show that $P_n+(n+1)^3=P_{n+1}$.
Note that $1+2+\cdots+n=n(n+1)/2$.
$$\eqalign{
P_n+(n+1)^3&=[n(n+1)/2]^2+(n+1)^3\cr
&=(n+1)^2[(n/2)^2+(n+1)]\cr
&=(n+1)^2(n^2/4+n+1)\cr
\cr
P_{n+1}&=[n(n+1)/2+(n+1)]^2\cr
&=[(n+1)(n/2+1)]^2\cr
&=(n+1)^2(n/2+1)^2\cr
&=(n+1)^2(n^2/4+n+1)\cr
}$$