Can my proof be simplified and is it valid? $n(n+2) = (n+1)^2 - 1$ for all integers I've decided to learn the basics of proofs and here is my first attempt. Could I improve or simplify my proof in any way? Is my formal language correct? Thanks!
Let $n$ be any integer. $$n(n+2) = (n+1)^2 - 1$$ I will prove this identify by induction.
First, check with $n=1$;
$$1 \times 3 = 2^2 + 1 \equiv
3 = 3$$
Inductive step: Assume that the identity is true for n = k;
$$k(k+2) = (k+1)^2 - 1$$
When $n = k + 1$;
$$(k+1)((k+1)+2) = ((k+1)+1)^2 - 1$$
$$\equiv k+1(k+3) = (k+2)^2 - 1$$
Let $n = k + 1$;
$$n(n+2) = (n+1)^2 - 1$$
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Using $a^2 - b^2 = (a-b)(a+b)$ we have $$(n+1)^2 - 1 = (n+1)^2 - 1^2 = (n+1-1)(n+1+1) = n(n+2).$$
