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 $2,2.2^r \geq 2.3 r 2^{r+1} \geq 6 r 2^{r+1} \geq 3 r+3 r 2^{r+1} \geq 3$ $+3 r$,
[Since $3 r \geq 3 \Rightarrow 3 r+3 r \geq 3+3 r] 2^{r+1} \geq 3 r(r+1) \Rightarrow P(r+1)$ is true.

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