i need to proof the following formula by math. induction
$ \sum_{k=1}^n k!*k=(n+1)!-1 $
n ∈ N
I dont know how to start and there to end, i would appreciate any help.
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This question seems to be very popular. – StubbornAtom Nov 05 '16 at 17:54
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2As for how to set up induction proofs in general, visit this question/answer for a general format. – JMoravitz Nov 05 '16 at 17:58
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@JMoravitz Thanks for that. – StubbornAtom Nov 05 '16 at 18:01
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Note that $kk!=(k+1)!-k!$. Now, evaluate the telescoping summation. – Mark Viola Nov 05 '16 at 18:03
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Basis step : $\sum_{k=1}^{1} k!*k = 1!*1 = 2!-1$
Induction step : $$\begin{align} \sum_{k=1}^n k!*k=(n+1)!-1\\ &\implies \sum_{k=1}^{n+1} k!*k\\ &= \sum_{k=1}^{n} k!*k + (n+1)!*(n+1)\\ &=(n+1)!-1 +(n+1)!*(n+1)\\ &=(n+1)!(n+1+1) -1\\ &=(n+2)! -1 \end{align}$$
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