Principle of Mathematical Induction - Practice Questions & MCQ

Mathematical induction is one of the techniques which can be used to prove a variety of mathematical statements which are formulated in terms of n, where n is a positive integer (natural number)

Principle of Mathematical Induction

The statement is true for $\mathrm{n}=1$, i.e., $\mathrm{P}(1)$ is true, and
If the statement is true for $\mathrm{n}=\mathrm{k}$ (where k is some positive integer), then the statement is also true for $\mathrm{n}=\mathrm{k}+1$, i.e., truth of $\mathrm{P}(\mathrm{k})$ implies the truth of $\mathrm{P}(\mathrm{k}+1)$.

Then, $\mathrm{P}(\mathrm{n})$ is true for all natural numbers n .
Property (i) is simply a statement of fact. There may be situations when a statement is true for all $\mathrm{n} \geq$ 2. In this case, step 1 will start from $\mathrm{n}=2$ and we shall verify the result for $\mathrm{n}=2$, i.e., $\mathrm{P}(2)$.

Property (ii) is a conditional property. It does not assert that the given statement is true for $\mathrm{n}=\mathrm{k}$, but only that if it is true for $\mathrm{n}=\mathrm{k}$, then should also be true for $\mathrm{n}=\mathrm{k}+1$ for this principle to be applicable. So, to prove that the property holds, only prove that conditional proposition:

If the statement is true for $n=k$, then it is also true for $n=k+1$.