Q.What do you mean by the inductive hypothesis?

The inductive hypothesis is the 'assume true for $n=k$' step used as a stepping stone to prove the statement for $n=k+1$.

The Principle of Mathematical Induction proves a statement $P(n)$ is true for all natural numbers $n$ in two steps:

1. Base step — verify $P(1)$ (or the smallest relevant $n$) is true directly.
2. Inductive step — show that IF $P(k)$ is true for some natural number $k$, THEN $P(k+1)$ must also be true.

The assumption made at the start of the inductive step — that $P(k)$ holds for some particular $k$ — is called the inductive hypothesis. It is not something we prove; it is a temporary assumption used purely as a tool to derive $P(k+1)$. Once both steps are established, the truth of $P(n)$ for every natural number $n$ follows automatically, like a chain of falling dominoes.

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