<s>[INST] 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
 [/INST] First, show that this is true for $n=5$:
$5^2+1<2^5$
Second, assume that this is true for $n$:
$n^2+1<2^n$
Third, prove that this is true for $n+1$:
$(n+1)^2+1=$
$n^2+\color\green{2}\cdot{n}+1+1<$
$n^2+\color\green{n}\cdot{n}+1+1=$
$n^2+n^2+1+1=$
$2\cdot(\color\red{n^2+1})<$
$2\cdot(\color\red{2^n})=$
$2^{n+1}$

Please note that the assumption is used only in the part marked red.
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