Q.Using the Mathematical induction, show that for any natural number $n\ge 2$,

[!FORMULA] $\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\dfrac{1}{1+2+3+4}+\cdots+\dfrac{1}{1+2+3+\cdots+n} = \dfrac{n-1}{n+1}.$

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, be …