posted by qwerty .
i know how to write/make explicit formulas but i have tried long and hard to find the formula for the sequence 2,7,15,26,40 i know that it increases by "3 more than the last increase" every time but i cant write a equation for it.
it describes the number of total cards of a cardhouse as each bottom layer is added
I don't know what method you use, but I noticed that the second differences for the data is a constant 3, as you noted.
That tells us that the data can be expressed by a quadratic function
let the function by y = ax^2 + bx + c
and sub in (1,2),(2,7), and (3,15), the first 3 given values.
this gives you 3 equations in 3 unknowns which solve quite nicely to
a = 3/2
b = 1/2
c = 0
so the function is y = (3/2)x^2 + x/2
y = (3x^2 + x)/2
test it for the last value, it works.