Question

Let P(n) be the statement : $2^n\le 3n$. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all $nϵN$.

Answer 1

P(n) : $2^n\ge 3n$

It is given that P(r) is true, so

$2^r\ge 3^r$ .......(1)

Multiplying both the sides by 2,

$2^r.2\ge 3r.2$

$2^{r+1}\ge 6r$

$2^{r+1}\ge 3r+3r$

$2^{r+1}\ge 3+3r$

[$Since3r\ge 3,6r\ge 3+3r$]

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

So, P(r + 1) is true

But for r = 1

$2\ge 3$

It is true, so

P(n) is not true for all $nϵN$ by PMI