Question

Prove that 3^n is greater than n^2 for all n greater or equal to 0. I know that this proof will be by induction

Answer 1

\[3^{n}>n ^{2},\forall n \in \mathbb{N} \]

Answer 2

hmmm its a while since I did these but here goes:

if n = 1 we have

3^1 > 1^2
3 > 1 which is evidently true

assuming the inequality is true then
if n = k then
3^k > k^2

if n = k+1 then
3^(k+1) > (k + 1)^2

so the inequality is true for n=k then its also true for n = k+1

we have shown thats its treu for n = 1
therefore it must be true for n =2 , n = 3 ,4,5 etc

so it must be true for all n>0