Question

A certain business has implemented a new incentive plan whereby employees earn tokens for every sale they make.

For 1 sale, an employee earns 1 tokens.
For 2 sales, an employee earns 6 tokens.
For 3 sales, an employee earns 12 tokens.
For 4 sales, an employee earns 19 tokens.

Which recursive equations represents the pattern?

Answer 1

For the answer to the question above,
Suppose that for x sales the number of tokens the employee earns is:
ax^3 + bx^2 + cx + d
Therefore:
a(1^3) + b(1^2) + c(1) + d = 1
a(2^3) + b(2^2) + c(2) + d = 6
a(3^3) + b(3^2) + c(3) + d = 12
a(4^3) + b(4^2) + c(4) + d = 19
Therefore:
a + b + c + d = 1
8a + 4b + 2c + d = 6
27a + 9b + 3c + d = 12
64a + 16b + 4c + d = 19
Solve that to get:
a = 0
b = 1/2
c = 7/2
d = -3
Therefore, the formula is:
(1/2)x^2 + (7/2)x - 3