<s>[INST] finding nth term Let 
  3,8,17,32,57 . . . . .
be a pattern.How do we find the nth number?My brains are completely jammed,I am tired.I do not even recognize the pattern.I calculated a few ways,but all I want is a little hint,not the whole solution.
 [/INST] If you want the smallest degree polynomial that gives these numbers (which is only one of the infinitely many possible explanations for them), we can use the calculus of finite differences:
$$\fbox{3}\quad 8\quad 17\quad 32\quad 57 \\
\fbox{5}\quad 9\quad 15\quad 25\\
\fbox{4}\quad 6\quad 10\\
\fbox{2} \quad 4\\
\fbox{2}$$
Define
$$\begin{align*}
f(n)&=3\binom{n}{0}
+5\binom{n}{1}+4\binom{n}{2}+2\binom{n}{3}+2\binom{n}{4}\\\\
&=3 + \frac{19 n}{6} + \frac{23 n^2}{12} - \frac{n^3}{6} + \frac{n^4}{12}
\end{align*}$$
Then
$$f(0)=3,\quad f(1)=8,\quad f(2)=17,\quad f(3)=32,\quad f(4)=57$$
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