How do I convert this mathematical progression into a formula?

Question

I have a mathematical progression that I came up with for a sequence of thresholds for a video game. It is calculated by adding three to the first term, four to the second term, 5 to the third term, and so on.

X | Y

------

1 | 3

2 | 7

3 | 12

4 | 18

5 | 25

6 | 33

7 | 42

How do I convert this to a formula so I can solve for X?

Answer 1

AN = (1/2)(N+5/2) - 25/8 for the Nth term

A1 = (1/2)(1+5/2)^2 - 25/8 = (1/2)7/2)^2 -25/8 = 49/8-25/8 = 24/8= 3

A2 =(1/2)(2+5/2)^2 -25/8 = (1/2)(9/2)^2 -25/8 = 81/8-25/8 = 56/8 = 7

...

A7 = (1/2)(7+5/2)^2 -25/8 = (1/2(19/2)^2 -25/8 = (361-25)/8 = 336/8 = 42

the relation is a parabolic function

y=a(x-h)^2 + k

fill in 3 points and get 3 equations, 3 unknowns, solve for a,h, and k

a = 1/2, h= -5/2, k= -25/8

vertex of the parabola is (-5/2,-25/8)