If P(n) is the statement " 2^n 3 n " and if P(r) is true, prove t… - Vidyadip

Question

If $P(n)$ is the statement " $2^n \geq 3 n$ " and if $P(r)$ is true, prove that $P(r+1)$ is true.

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Answer

$P(n): 2^n \geq 3 n$
Given that $P(r)$ is true
$\Rightarrow 2^r \geq 3 r$
Multiplying both sides by $2$ ,
$2.2^r \geq 2.3 r$
$2^{r+1} \geq 6 r$
$2^{r+1} \geq 3 r+3 r$
$\left.2^{r+1} \geq 3+3 r, \text { [Since } 3 r \geq 3 \Rightarrow 3 r+3 r \geq 3+3 r\right]$
$2^{r+1} \geq 3 r(r+1)$
$\Rightarrow P(r+1) \text { is true. }$

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