Question

For all positive integers $n$, which among the following is true?

• A. $2^n<n$
• B. $2^n>n$
• C. $2^n=n$
• D. None of these

Answer 1

Answer: (B)

Let $P(n):2^n>n$
When $n=1,2^1>1$. Hence, $P(1)$ is true.
Assume that $P(k)$ is true for some positive integer $k$, i.e.,
$2^k>k$
We shall now prove that $P(k+1)$ is true whenever $P(k)$ is true. Multiplying both sides of Eq. (i) by 2, we get
$2\cdot 2^k>2k$
i.e., $2^{k+1}>2k=k+k>k+1$
Therefore, $P(k+1)$ is true when $P(k)$ is true. Hence, by principle of mathematical induction, $P(n)$ is true for every positive integer $n$.