I'm currently working on this problem which includes adding and multiplying (and probably factoring) factorials:
$(k+1)!-1+(k+1)(k+1)!=(k+2)!-1$
I need to make the left hand side equal to the right hand side without manipulating the right hand side. I've tried looking up methods on how to do this, but every example I've been able to find only showed cases where the denominator and numerator had factorials.
I know that $(k+1)!$ has to factor somehow, but I honestly don't know how to do it with factorials in this case.
Factor the $(k+1)!$ from the LHS:
LHS=
$(k+1)!(1+k+1)-1$
=$(k+1)!(k+2)-1$
=$(k+2)!-1$$\quad$ {because $(k+2)! = (k+1)!(k+2)$}
= RHS
QED