Question

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

help

Answer 1

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
or
y = (3x^2 + x)/2

test it for the last value, it works.