What is the nth term rule of the quadratic sequence below?

-4, - 3, 0, 5, 12, 21, 32

The nth term = n^2 - 2n - 3.

Step-by-step explanation:

-4 - 3 0 5 12 21 32

Differences 1 3 5 7 9 11

Difference 2 2 2 2 2 <- - divide this by 2 so its n^2.

So the first term is n^2

List the original numbers and values of n^2:

sequence - 4 - 3 0 5 12 21 32

n^2 1 4 9 16 25 36 49

Subtract - 5 - 7 - 9 - 11 - 13 - 15 - 17

These last numbers have a common difference of - 2 so the next part of the answer is - 2n and get the - 5, - 7 etc we add - 3.

So our formula is n^2 - 2n - 3.

Check for the 3rd term:

3rd term is 3^2 - 2 (3) - 3 = 0.

5th term = 5^2 - 2 (5) - 3 = 25 - 10 - 3 = 12.