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Q.
For all positive integers $n$, which among the following is true?
Principle of Mathematical Induction
Solution:
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 >2 k$
i.e., $ 2^{k+1} >2 k =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 $n$.