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.

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
$$f(0)=3,\quad f(1)=8,\quad f(2)=17,\quad f(3)=32,\quad f(4)=57$$
