# Thread: simple yet clever proof with factorials

1. ## simple yet clever proof with factorials

we're supposed to use the principle of mathematical induction for this.

basically the inductive part boils down to proving algebraically that
$(k+1)!-1+(k+1)\cdot(k+1)!=(k+2)!-1$

but i'm not familiar enough with factorials to show this. i've read about factorials on wikipedia but they didn't have any good identities or anything i could use. any suggestions would be very helpful but for now i'm going to just stare at it for a few minutes and read about series with factorials.

thanks guys

2. Originally Posted by hairy
we're supposed to use the principle of mathematical induction for this.

basically the inductive part boils down to proving algebraically that
$(k+1)!-1+(k+1)\cdot(k+1)!=(k+2)!-1$

but i'm not familiar enough with factorials to show this. i've read about factorials on wikipedia but they didn't have any good identities or anything i could use. any suggestions would be very helpful but for now i'm going to just stare at it for a few minutes and read about series with factorials.

thanks guys

If you already realized that $n!=n(n-1)!$ then it is almost trivial

$(k+1)!-1+(k+1)\cdot(k+1)!=(k+1)!\left[1+k+1\right]-1=(k+1)!(k+2)-1=...$

3. Originally Posted by tonio
If you already realized that $n!=n(n-1)!$ then it is almost trivial

$(k+1)!-1+(k+1)\cdot(k+1)!=(k+1)!\left[1+k+1\right]-1=(k+1)!(k+2)-1=...$

thanks, i see.