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I'm working on a practice set:

Solve the recurrence $a_n=na_{n−1}+n!$ for $n>0$ with $a_0=1$ Give a simple expression for $a_n$

For this problem I know the answer is $(n+1)!$ But I'm not sure how to get there....

Here is what I did so far:

I divided the equation by n so: $\frac{a_n}{n} = a_{n-1} + (n-1)!$

Then I used telescoping, so: $\frac{a_n}{n} = a_{n-1} + (n-1)!$

$\frac{a_{n-1}}{n-1} = a_{n-2} + (n-2)!$

$\frac{a_{n-2}}{n-2} = a_{n-3} + (n-3)!$

$...$

So I cancel terms across the equal sign and I get:

$a_n = a_0 + (n-1)!$

But this is not correct.

Thanks for help

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    $\begingroup$ Did you try proving it by induction? $\endgroup$ Feb 16 '20 at 20:05
  • $\begingroup$ I haven't.... the assignment was on telescoping so was trying to first solve it that way - but open to any info if you sense you can use induction:) $\endgroup$ Feb 16 '20 at 20:08
  • $\begingroup$ If you edit in your telescoping argument, we may identify the mistake. $\endgroup$
    – J.G.
    Feb 16 '20 at 20:11
  • $\begingroup$ Hi! Thanks so much @J.G. I added my incorrect telescoping process $\endgroup$ Feb 16 '20 at 20:19
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Telescope $b_n:=a_n/n!$, which satisfies $b_n-b_{n-1}=1$ because$$b_n=\frac{na_{n-1}+n!}{n!}=\frac{a_{n-1}}{(n-1)!}+1=b_{n-1}+1.$$

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  • $\begingroup$ +1; this works well, and I think it's what OP was looking for $\endgroup$ Feb 16 '20 at 20:13
  • $\begingroup$ @J.G. would like to understand a bit more about your process. I'm not sure I understand $b_n := a_n/n!$ curious how you got there - thanks! $\endgroup$ Feb 16 '20 at 20:20
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    $\begingroup$ @KatieMelosto I divide by something that multiplies by $n$ at each step, because you need to telescope something of the form $b_n-b_{n-1}$. Your mistake was in trying to telescope something that isn't of that form. $\endgroup$
    – J.G.
    Feb 16 '20 at 20:32
  • $\begingroup$ @J.G.Thanks so much! When you divide $b_n := a_n/n!$ is that: $(a_n = na_{n-1} + n!)/n!$? Trying to put the math together $\endgroup$ Feb 16 '20 at 20:43
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    $\begingroup$ @KatieMelosto See edit. $\endgroup$
    – J.G.
    Feb 16 '20 at 21:25
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Induction works:

base case: $n=0$

induction step: $a_{n+1}=(n+1)a_n+(n+1)!=(n+1)(n+1)!+(n+1)!=(n+2)!$

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