Prove that n^2+n is even

To Prove

n2 + n is even ∀ n ∈ Z

Proof

n2 + n = n * (n + 1)

For the value of n, we have 2 cases

  1. n is even
    If n is even, then n * (n + 1) will also be even since multiplying even number with any integer gives even number.
  2. n is odd
    If n is odd, then (n + 1) is even since adding odd number with odd number gives even number. This implies that n * (n + 1) will also be even since multiplying even number with odd number gives even number.

Therefore, we can say that n2 + n is even ∀ n ∈ Z.

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