Question

Prove $3^{n+1}>$$3(n+1)$

Answer 1

Let P(n): $3^{n+1}>3(n+1)$
Step 1: Put $n=1$
Then, $3^2>3\left(2\right)$
$⟹$ p(n) is true for $n=1$
Step 2: Assume that P(n) is true for $n=k$
Then, $3^{k+1}>3(k+1)$
Multiplying throughout with '3'
$3^{k+1}\times 3>3(k+1)\times 3=9k+9=3(k+2)+(6k+3)>3(k+2)$
$⟹3^{\overline{k+1}+1}>3(\overline{k+1}+1)$
P(n) is true for $n=k+1$
$\therefore$ By the principle of mathematical induction, P(n) is true for all $n\in N$
Hence, $3^{n+1}>3(n+1)$.