Question

Find the first five terms of the quadratic polynomial sequence with the formula a^n=3n^2-2. Calculate the 1st differences. Then, calculate the 2nd differences and show that they are constant. Include the constant value for the 2nd differences in your response.

Answer 1

To find the first five terms of the quadratic polynomial sequence, we substitute n = 1, 2, 3, 4, 5 into the formula a^n = 3n^2 - 2:

a^1 = 3(1)^2 - 2 = 3 - 2 = 1
a^2 = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10
a^3 = 3(3)^2 - 2 = 3(9) - 2 = 27 - 2 = 25
a^4 = 3(4)^2 - 2 = 3(16) - 2 = 48 - 2 = 46
a^5 = 3(5)^2 - 2 = 3(25) - 2 = 75 - 2 = 73

The first five terms of the sequence are 1, 10, 25, 46, 73.

To calculate the 1st differences, we find the difference between consecutive terms:

10 - 1 = 9
25 - 10 = 15
46 - 25 = 21
73 - 46 = 27

The 1st differences are 9, 15, 21, 27.

To calculate the 2nd differences, we find the difference between consecutive 1st differences:

15 - 9 = 6
21 - 15 = 6
27 - 21 = 6

The 2nd differences are all 6.

Since the 2nd differences are constant, the sequence is a quadratic polynomial sequence.