Question

Let $P\left(n\right)$ be the statement $3^n>n^n$. If $P\left(n\right)$ is true, prove that $P(n+1)$ is true.

Answer 1

$P\left(n\right):3^n>n^n$

for $n=1$

$P\left(1\right):3^{\prime }1>1^1$

$3>1$

$\therefore$ It is true for $n=1$

$P\left(K\right)\Rightarrow 3^k>k^k$

It is given that $P\left(h\right)$ is true, so $P\left(k\right)$ is true for $n=k$

$P(k+1)=3^{k+1}>(k+1{)}^{k+1}$

$3^k{.3}^1>(k+1{)}^k(k+1)$

By the principle for mathematics production,

$P(n+1)$ is true, when $P\left(n\right)$ is true

$\therefore P(n+1)$ is true.