Proof by induction: $3^n > n^2$ for all integers greater or equal to 1

Question

I have to prove that $3^n\geq n^2$ for all integers $n\geq1$ .

This task is quite similar to other like the one given in 1, but with the difference that it has to be also valid for $n=1$. The usual approach uses $2n^2 > n^2 + 2n + 1$, which is just not true for $n=1$ as $n^2\ngeq 2n + 1$.

Are there any suggestions on how to approach this?

What do you have to prove? Your title says "$2^n>n^2$", but your question says "$3^n\geq n^2$". — Ben Sheller, Oct 14 '15 at 16:55

score 1 · Accepted Answer · answered Oct 14 '15 at 16:54

1

It is true for $n=1$ and $n=2$. Now start your induction for $n \geq 2$.

Above answer is for case $2^n$. For case $3^n$ we actually have $3^n>2^n$.

answered Oct 14 '15 at 16:54

vudu vucu

1,054

Proof by induction: $3^n > n^2$ for all integers greater or equal to 1

1 Answers1