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A sequence of numbers $$a_1, a_2,a_3,a_4$$... is defined by

$$a_1=1 \cdot 1$$

$$a_2=1 \cdot 2 + 2\cdot1$$

$$a_3=1\cdot3+2\cdot2+4\cdot1$$

$$a_4=1\cdot4+2\cdot3+4\cdot2+8\cdot1$$

...

(a) Use the pattern to write out an expression for $$a_5$$, then compute its value.

(b) There is a short formula for $$a_n$$ not involving a sum. Use whatever mathematical techniques you know to find or guess the formula.

(c) If you are able to, prove that your formula works for all $$a_n$$.

I already got the answers to (a) and (b), I just need help with (c).

The answer to (a) is 57, and the answer to (b) is $$a_n=2^{n+1}-(n+2)$$. I just have no idea why this formula works.