Using Induction

Aquafina · #1 $Old$ November 2nd, 2009, 07:05 AM

I couldn’t do the induction steps for these questions. Perhaps induction is not the best method for them:

1. Prove that for n E Z ,n>2, n^(n+1)>(n+1)^n

2. Prove that 1!+ 2!+ 3!+... has no square values for n>3

3. Show that n^5-n^3 is divisible by 12

Bruno J. · #2 $Old$ November 4th, 2009, 11:31 PM

Hello!

1) Might be easier if you rearrange it as $n>\left(1+\frac{1}{n}\right)^n$ ...

2) $n! \equiv 0 \mod 4$ for $n>3$ ; so $1!+2!+3!+...+n! \equiv 1!+2!+3! \equiv 3 \mod 4$ for $n>3$ , but no square is congruent to 3 modulo 4.

3) $n^5-n^3=n^3(n^2-1)=n^3(n-1)(n+1)$ . Show that this is always divisible by 3 and always divisible by 4.