Show by induction that $(n^2) + 1 < 2^n$ for intergers $n > 4$ So I know it's true for $n = 5$ and assumed true for some $n = k$ where $k$ is an interger greater than or equal to $5$.
for $n = k + 1$ I get into a bit of a kerfuffle.
I get down to $(k+1)^2 + 1 < 2^k + 2^k$ or equivalently:
$(k + 1)^2 + 1 < 2^k * 2$.
A bit stuck at how to proceed at this point

$\displaystyle
2^n = (1+1)^n > \binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}>n^2+1
$
iff $n>5$. (*)
The case $n=5$ is proved by inspection.
(*) This seems to be a cubic inequality but it reduces to an easy quadratic.
