<s>[INST] Prove by induction that for every $n \ge 0$ the term $2^{2n+1} - 9n^2 + 3n -2$ is divisible by $54$ What I get is $4 \cdot 54k + 27(n(n-1)) - 2$.
I understand first two terms but $-2$ is giving me a headache. I suppose to be true every term should be divisible by $54$. Or did I make wrong conclusion?
P. S: Is there mathjax tutorial?
 [/INST] Let $f(n) = 2^{2n+1}-9n^2+3n-2$; then
\begin{align*}
  f(n+1)-4f(n) &= 2^{2n+3}-9(n+1)^2+3(n+1)-2-4(2^{2n+1}-9n^2+3n-2) \\
      &= 4\cdot 2^{2n+1} - 9n^2 - 18n - 9 + 3n+3-2-4(2^{2n+1}-9n^2+3n-2) \\
      &= 27n^2-27n  = 27n(n-1).
\end{align*}
Since either $n$ or $n-1$ is even, this expression is divisible by $54$. Since $4f(n)$ is divisible by $54$ (inductive assumption), so is $f(n+1)$.
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