Prove that `3^n>n^2`

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Let P(n) : `3^n>n^2` be statement ,

To prove P(n) is true for all natural number . We prove it by princple of mathematical induction.

P(1) : `3^1 >1^2=1` which is true.

Let P(k) is true .

P(k) : `3^k > k^2` is true.

To prove P(k+1) is true when P(k) is true.

P(k+1) : `3^(k+1) > (k+1)^2`

`since`

`3^k>k^2`

`3^k>1`

`and 3^k>2k`

`therefore`

`3^k+3^k+3^k>k^2+1+2k`

`3.3^k>(k+1)^2`

`therefore`

`3^(k+1)>(k+1)^2`

`Thus`

`` P(k+1) is true when P(k) is true

Thus P(n) is true for al natural number.

Hence proved.

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