Question

Use induction to prove that 2? ?? for any integer n>0 . Indicate type of induction used.

Answer 1

Answer with explanation:

The given statement is which we have to prove by the principal of Mathematical Induction

    $2^{n}>n$

1.→For, n=1

L H S =2

R H S=1

2>1

L H S> R H S

So,the Statement is true for , n=1.

2.⇒Let the statement is true for, n=k.

      $2^{k}>k$

                   ---------------------------------------(1)

3⇒Now, we will prove that the mathematical statement  is true for, n=k+1.

     $\rightarrow 2^{k+1}>k+1\\\\L H S=\rightarrow 2^{k+1}=2^{k}\times 2\\\\\text{Using 1}\\\\2^{k}>k\\\\\text{Multiplying both sides by 2}\\\\2^{k+1}>2k\\\\As, 2 k=k+k,\text{Which will be always greater than }k+1.\\\\\rightarrow 2 k>k+1\\\\\rightarrow2^{k+1}>k+1$

Hence it is true for, n=k+1.

So,we have proved the statement with the help of mathematical Induction, which is

      $2^{k}>k$