Q.Using the principle of mathematical induction prove that $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots + \dfrac{1}{2^n} = 1 - \dfrac{1}{2^n}$.

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 …