Q.By the principle of mathematical induction, prove that, for $n\ge 1$

[!FORMULA] $1^3+2^3+3^3+\cdots+n^3 = \left(\dfrac{n(n+1)}{2}\right)^2$

The Principle of Mathematical Induction (PMI) is a method for proving that a statement $P(n)$, framed for every natural number $n$, is true for all $n \in \mathbb{N}$ at once — without needing to verify infinitely many individual cases directly. It rests on two pillars: a base case, where $P(1)$ (or the smallest relevant value of $n$) is checked directly, and an inductive step, where it is shown that the truth of $P(k)$ forces the truth of $P(k+1)$ for every positive integer $k$. Once both are established, PMI concludes that $P(n)$ holds for every natural number, because the chain of implications $P(1) \Rightarrow P(2) \Rightarrow P(3) \Rightarrow \cdots$ never breaks. It underlies proofs of summation formulas, divisibility results, inequalities, and identities involving powers and products throughout algebra.