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

1 Answer

2 votes

Answer:

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).

User Yazid Mekhtoub
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