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.