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

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

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

Download App

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.

✓

Answer

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

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Mathematical Induction→Mathematical Induction — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Mathematical Induction — Maths STD 11 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions