<s>[INST] Prove the inequality using induction Prove $3^n \geq 2n^2 +1$ for $n = 1,2,\ldots$ using induction
This is what I have so far
Base case - $n=1,3^1 \geq 2^1 + 1 = 3$ true
Induction step - Assume true for some n, then,
$ 3*3^n \geq 3*(2n^2 +1)=6n^2 +3 $ 
I have to somehow manipulate and show its $\geq 2*(n+1)^2 +1$ 
 [/INST] First, show that this is true for $n=1$:
$3^1\geq2\cdot1^2+1$
Second, assume that this is true for $n$:
$3^n\geq2n^2+1$
Third, prove that this is true for $n+1$:
$3^{n+1}=$
$3\cdot\color\red{3^n}\geq$
$3\cdot(\color\red{2n^2+1})=$
$6n^2+3=$
$2n^2+4n^2+3\geq$
$2n^2+4n+3=$
$2n^2+4n+2+1=$
$2(n^2+2n+1)+1=$
$2(n+1)^2+1$

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