Question

Prove that $2^n>n$ for all positive integers $n$.

Answer 1

Step $\left(1\right):$ Assume given statement
Let the given statement be $P\left(n\right)$, i.e.
$P\left(n\right):2^n>n$

Step $\left(2\right):$ Checking statement $P\left(n\right)$ for $n=1$
Put $n=1$ in $P\left(n\right)$, we get
$P\left(1\right):2^1>1$
$\Rightarrow 2>1$, which is true
Thus $P\left(n\right)$ is true for $n=1$.

Step $\left(3\right):$ $P\left(n\right)$ for $n=k$.
Put $n=K$ in $P\left(n\right)$, and assume that $P\left(K\right)$ is true for some positive integer $K$ i.e.,
$P\left(K\right):2^K>K\cdots \left(1\right)$

Step $\left(4\right):$ Checking statement $P\left(n\right)$ for $n=K+1$
Now, we shall prove that $P(K+1)$ is true whenever $P\left(k\right)$ is true.
Multiply both sides of $\left(1\right)$ by $2$, we get
$2\cdot 2^K>2K$ i.e.,
$2^{K+1}>K+K$
$2^{K+1}>K+1$
$(\because K+K>K+1)$
Thus, $P(K+1)$ is true whenever $P\left(K\right)$ is true.
Final Answer:
Hence, by principle of Mathematical Induction, $P\left(n\right)$ is true for every positive integer $n$.