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?

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