`-1, 8, 23, 44, 71, 104....` Decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the...

The given sequence is:

`-1, 8 , 23, 44, 71, 104`

To determine if it is a linear sequence, take the difference between consecutive terms.

`-1, 8 , 23, 44, 71, 104`

    `vvv` `vvv` `vvv`  `vvv`  `vvv`

    `9` `15``21` `27` `33`

Notice that the result are not the same. So the sequence is not linear.  

Then, take the second difference of the consecutive terms to determine if it is quadratic.

`9, 15, 21, 27, 33`

 `vvv`  `vvv`   `vvv`  `vvv`

 `6`   `6`   `6`   `6`

Since the second difference of the consecutive terms are the same, the sequence is quadratic.

To determine the model of the quadratic expression apply the formula:

`T_n = an^2 + bn + c`

where Tn is the nth term of the sequence.

To solve for the values of a, b and c,  consider the first few three terms of the sequence.

Plug-in T1=-1 and n=1.

`-1=a(1)^2+b(1)+c`

`-1=a+b+c `              (Let this be EQ1.)

Also, plug-in T2=8 and n=2.

`8=a(2)^2 + b(2) + c`

`8=4a+2b+c`           (Let this be EQ2.)

And, plug-in T3=23 and n=3.

`23=a(3)^2+b(3)+c`

`23=9a+3b+c `        (Let this be EQ3.)

Then, apply elimination method of system of equations. Let's eliminate variable c. 

To eliminate c, subtract EQ2 from EQ1.

EQ1:        `-1=a+b+c`

EQ2:      `-(8=4a+2b+c)`

`-----------------`

               `-9 = -3a -b`

And simplify the resulting equation.

`-9=-3a-b`

`9=3a+b `        (Let this be EQ4.)

Eliminate c again. So subtract EQ3 from EQ1.

EQ1:      `-1=a+b+c`

EQ2:  `-(23=9a+3b+c)`

`-----------------`

           `-24=-8a-2b`

And, simplify the resulting equation.

`-24=-8a-2b`

`12=4a+b`        (Let this be EQ5.)

Then, eliminate another variable. Let it be c.

To eliminate c, subtract EQ4 from EQ5.

EQ5:    `12=4a+b`

EQ4:   `-(9=3a+b)`

`--------------`

              `3=a`

Now that the value of a is known, plug-in it to either EQ4 or EQ5. Let's use EQ4.

`9=3a+b`

`9=3(3)+b`

`0=b`

Then, plug-in the value of a and b to either EQ1, EQ2 or EQ3. Let's use EQ1.

`-1=a+b+c`

`-1=3+0+c`

`-4=c`

Now that values of a, b and c, plug-in them to the formula of nth term of quadratic expression.

`T_n=an^2+bn+c`

`T_n=3n^2+(0)n+(-4)`

`T_n=3n^2-4`

Therefore, the given -1, 8 , 23, 44, 71, 104 is a quadratic sequence. The model for its nth term is `T_n=3n^2-4` .

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