Question

Show that, for any integer n > 2, (n + 1)" - 1 is divisible by n2. (Hint: Use the Binomial Theorem.)

Answer 1

It's true

Explanation:

Operating the square we have:

`(n+1)^(2) -1= (n^(2) +2n+1)-1\\ n^(2) +2n+1-1= n^(2) +2n`

Here we can factorize n:

`n^(2) +2n=n*(n+2)`

The last line means our number (
`(n+1)^(2) -1`) is divisible by n.

The clause of n>2 is true, but even its true for
`n\geq 0` because when n=1 the theorem means that the expression is divisible by 1, but this it's true for every integer, and for n=0 the expression will be 0 and 0 it´s divisible by 0 (Following the definition a|b if a=nb, with a, b and n integers).