Let P(n) be the statement :

Question:

Let P(n) be the statement : 2n ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all n ∈ N?

Solution:

$P(n): 2^{n} \geq 3 n$

We know that $P(r)$ is true.

Th $u s$, we have :

$2^{r} \geq 3 r$

To show: $P(r+1)$ is true.

We know :

$P(r)$ is true.

$\therefore 2^{r} \geq 3 r$

$\Rightarrow 2^{r} .2 \geq 3 r .2 \quad$ [Multiplying both sides by 2]

$\Rightarrow 2^{r+1} \geq 6 r$

$\Rightarrow 2^{r+1} \geq 3 r+3 r$

$=2^{r+1} \geq 3 r+3 \quad[$ Since $3 r \geq 3$ for all $r \in N]$

$=2^{r+1} \geq 3(r+1)$

Hence, $P(r+1)$ is true.

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

$P(1): 2^{1} \ngtr 3.1$

Therefore, $P(\mathrm{n})$ is not true for all $n \in N$.