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

Question

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

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Answer

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

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