Question

Let the statement $P\left(n\right):2^n\ge 3n$, the truth of $P\left(k\right),\forall k\in N\Rightarrow$ truth of,

• A. $P\left(1\right)$
• B. $P\left(2\right)$
• C. $P\left(3\right)$
• D. $P(k+1)$

Answer 1

The correct option is D $P(k+1)$

Given $P\left(n\right):2^n\ge 3n$ .....(1)

Consider $P(k+1)$

$\Rightarrow 2^{k+1}\ge 3k+3$

$\Rightarrow 2^k.2\ge 3k+3$

$\Rightarrow 2^k\ge \frac{3k+3}{2}\ge 3k$

Hence, $2^k\ge 3k$

$\text{Truth of}P(k+1)\Rightarrow \text{Truth of P(k)}$
Also, we will check for $P\left(1\right),P\left(2\right),P\left(3\right)$

$P\left(1\right):2\ngeqq 3$

So, $P\left(1\right)$ is not true
$P\left(2\right):4\ngeqq 6$

So,$P\left(2\right)$ is not true
$P\left(3\right):8\ngeqq 9$

So,$P\left(3\right)$ is not true