Question

For all positive integers $n$, $P\left(n\right)$ is true , and $2^{n-2}>3n$, then which of the following is true?

• A. $P\left(3\right)$ is true.
• B. $P\left(5\right)$ is true.
• C. If $P\left(m\right)$ is true then $P(m+1)$ is also true.
• D. If $P\left(m\right)$ is true then $P(m+1)$ is not true.

Answer 1

Answer: (C)

If $P\left(m\right)$ is true then $P(m+1)$ is also true.

$2^{n-2}>3n$
Put $n=3$
$2\ngtr 6$
Hence, P(3) is not true.
Put $n=5$
$8\ngtr 15$
Hence, P(3) is not true.
Let $P\left(m\right)$ is true i.e.
$2^{m-2}>3m$
Now, we will check for $P(m+1)$
Consider , $2^{m-1}$
$=2^{m-2}.2$
$>2\left(3m\right)=6m=3m+3m$
$>3m+3$
Hence, $2^{m-1}>3(m+1)$
Hence, $P(m+1)$ is true.