Precalculus, Chapter 9, 9.4, Section 9.4, Problem 58

Firstly we need to determine whether the series is linear or quadratic. A linear sequence is a sequence of numbers in which there is a first difference between any consecutive terms is constant. However, a quadratic sequence is a sequence of numbers in which there is a second difference between any consecutive terms is constant.
Let's begin by determining if we have a first difference:
x_1 = T_2 - T_1 = 6 - 0 = 6
x_2 = T_3 - T_2 = 16 - 6 = 10
x_3 = T_4 - T_3 = 30 - 16 = 14
Clearly from above, we observe that we do not have a constant first difference, now let's observe the second difference by calculating it:
x_2 - x_1 = 10 - 6 = 4
x_3 - x_2 = 14 - 10 = 4
We observe from above we have a constant second difference.
Therefore the sequence is a quadratic model.
Now let's find the model:
A quadratic model is represented as follows:
T_n = an^2 + bn + c
Where: T_n = Term value, n= term number, variables: a,b,c
We need to determine the variables a, b and c to obtain our quadratic model. This is obtained as follows:
2a = second difference
2a = 4
a = 2
Now to determ6ine the variable b:
3a+ b = first difference between term 2 and term 1
3(2) + b = 6 (substitute 2 for a and 6 for T_2 - T_1)
b = 6 -6 =0
Lastly the variable c:
a + b + c = value of term 1
2 + 0 + c = 0 (substituted for a, b and value for term 1)
c = -2
Now we can develop our model:
T_n = 2n^2 -2
Let's double check the above model for term 2 and 5:
T_2 = 2(2)^2 -2 = 6
T_5 = 2(5)^2 - 2 = 48
SUMMARY:
Type of Sequence: Quadratic
Model:
T_n = 2n^2 -2