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: 

Clearly from above, we observe that we do not have a constant first difference, now let's observe the second difference by calculating it: 

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: 

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: 

 second difference

Now to determ6ine the variable b: 

 first difference between term 2 and term 1 

 (substitute 2 for a and 6 for T_2 - T_1)

 

Lastly the variable c: 

 value of term 1

 (substituted for a, b and value for term 1)

Now we can develop our model: 

Let's double check the above model for term 2 and 5:

SUMMARY:

Type of Sequence: Quadratic

Model: