How do you prove sequence results algebraically

The triangle number sequence starts 1 3 6 10
The nth term is given as a formula: 1/2n(n+1)
When you add two consecutive triangle numbers you always get a square number.
Prove this result algebraically..

Please help? If you need extra information please ask :)

If we have the nth triangle number, it is given by:

(1/2)n(n + 1)

The next triangle number is the (n + 1)th, which is:

(1/2)(n + 1)(n + 2)

Adding together:

(1/2)n(n + 1) + (1/2)(n + 1)(n + 2)
= (1/2)(n + 1)(n + n + 2) ... (factoring out (1/2)(n + 1))
= (1/2)(n + 1)(2n + 2)
= (1/2)(n + 1)2(n + 1)
= (n + 1)^2

Hope that helps!

Tn+Tn+1=n(n+1)/2 + (n+1)(n+2)/2 = (n+1)(n+n+2)/2= (n+1)(2n+2)/2= (n+1)^2 .