This question has three parts and tests mathematical induction.

$($i$)$ Give a recursive definition of the set $P$ of all positive integers greater than $0,$

$($ii$)$ formulate the applicable induction principle and then

$($iii$)$ apply the induction principle to prove that, for all integers ninN.

$1^3+2^3+$dots$+n^3=\frac{n^2(n+1{)}^2}{4}$

Note that this may also be written as

${\displaystyle \underset{i=1}{\overset{n}{}}}{i}^3=\frac{n^2(n+1{)}^2}{4}$