Quadratic Sequencing question help (not just the answer)?

Question

I am in a geometry class and were doing algebra review but i can't remember learning this. can someone help explain how to get the answer. (hopefully in steps)

Find the n'th term

0,5,12,21,...,n...

and

1,3,6,10,15,...,...n,

Answer 1

The first is quadratic since the second differences are all 2.  If you double each number and look for two things to multiply to get those numbers, you'll see this:

(skip 0 for a bit)

10 = 5•2

24 = 6•4

42 = 7•6

so you see the 0 is 4•0 to fit the pattern.  

If 0 is the first term (n=1), the formula then is (n+3)[2(n-1)]

That is (n+3)(2n - 2) or 2n^2 + 4n - 6.  But this was double the term so divide by 2 to get the answer: n^2 + 2n - 3

And the second is quadratic since second differences are 1.  Try the same trick:

2 = 2•1

6 = 3•2

12 = 4•3

20 = 5•4

so each is (n+1)(n) or n^2 + n, and then divide by 2 to get (n^2 + n)/2