Let P(n) be the statement:
Question:

Let $\mathrm{P}(n)$ be the statement $: 2^{n} \geq 3 n$. If $\mathrm{P}(r)$ is true, then show that $\mathrm{P}(r+1)$ is true. Do you conclude that $\mathrm{P}(n)$ is true for all $n \in \mathbf{N}$ ?

Solution:

Since, for $n=1$ i.e. P $(1)$ :

LHS $=2^{1}=2$

RHS $=3 \times 1=3$

As, LHS $<$ RHS

So, it is not true for $n=1$.

Hence, we conclude that $\mathrm{P}(n)$ is not true for all $n \in \mathbf{N}$.

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