Let $S(n)=1^3+2^3+$dots$+n^3.$ Prove by induction that AAn$1,S(n)=\frac{n^2(n+1{)}^2}{4}.$

Basis: $?$

Inductive Hypothesis: Assume that $S(n)=\frac{n^2(n+1{)}^2}{4}$ for any $n.$

Nothing to do here!

Prove that $S(n+1)=\frac{(n+1{)}^2(n+2{)}^2}{4}.$

$($Question: Which type of induction is used here?$)$