Triangle numbers are formed from the expression: 1/2n(n+1). Prove that the ratio between two consecutive triangle numbers is always n:n+2?

So if you pick some triangle number A = 1/2n(n+1), a consecutive triangle number would be B= 1/2(n+1)(n+2). Note that B is smaller than A as it has a larger denominator so when it comes to triangle numbers in order, it goes B, then A. If we put these numbers in ascending order into a ratio, we get: 1/2(n+1)(n+2):1/2n(n+1) Now if we multiply both sides by 2n(n+1), we can get rid of the denominator on the right side, which leaves us with: n/(n+2):1 Then we can multiply both sides by n+2 to get rid of the denominator on the left side, which gives us the ratio: n:n+2 Hope that makes sense :) ~ Sohini

Sohini Khan

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