I'm having a bit of trouble understanding mathematical induction, particularly when there's a question with powers or factorials. For example I have a problem 1 x 1! +2 x 2! + 3 x 3! +... + n x n! = (n+1)!-1.
I got (k+1)! ((k+1)-1) which I am unsure of. Can someone please explain to me.
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Most likely you're making a subtle shift in indices. Your induction step should look something like $$ \sum_{k=1}^{n-1} k \cdot k! = n! - 1 $$ at which point you add $n \cdot n!$ (the next term) to both sides. If you group things properly...
The key is of course to make sure that you are using the correct start and end points in your summation.
The base case is quickly checked: $1\times 1!=1=(1+1)!-1$
Now suppose this relation holds for $n$ and let's check $n+1$.
$$1\times 1!+2\times 2!+...+n\times n!+(n+1)\times(n+1)!=(n+1)!-1+(n+1)\times(n+1)!$$ Because we know that $1\times 1!+2\times 2!+...+n\times n!=(n+1)!-1$
Rearranging $(n+1)!-1+(n+1)\times(n+1)!$ we get $(n+1)!(n+1+1)-1$ but $(n+1)!(n+1+1)-1=(n+2)!-1$, as we wanted to show