`0, 6, 16, 30, 48, 70...` Decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model.

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`