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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. 

PLEASE HELP!!!!! THANKS!!!!

 Feb 23, 2020
 #1
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(a) I'm getting a_5 = 1*5 + 2*4 + 4*3 + 8*2 + 16*1 = 57.

 Feb 23, 2020

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