`a_1 = 0, a_2 = 8, a_4 = 30` Find a quadratic model for the sequence with the indicated terms.

The given sequence is:

`a_1 = 0` ,   `a_2=8` ,   `a_4=30`

To determine its quadratic model, apply the formula

`f(n) = an^2 + bn + c`

where f(n) represents the nth term of the sequence, `f(n)=a_n` .

So, plug-in the first term of the sequence.

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

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

Plug-in too the second term of the sequence.

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

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

And, plug-in the 4th term of the sequence.

`30=a(4)^2+b(4)+c`

`30=16a+4b+c `        (Let this be EQ3.)

To solve for the values of a, b and c, apply elimination method of system of equations.  In this method, a variable or variables should be removed.

Let's eliminate c. To do so, subtract EQ1 from EQ2.

EQ2:       `8=4a+2b+c`

EQ1:   `-(0=a+b+c)`

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

               `8=3a+b`          (Let this be EQ4.)

Let's eliminate c again. This time, subtract EQ2 from EQ3.

EQ3:     `30=16a+4b+c`

EQ2:   `-(8=4a+2b+c)`

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

                `22=12a+2b`

And this simplifies to:

`22/2=(12a+2b)/2`

`11=6a+b `          (Let this be EQ5.)

Then, eliminate b. To do so, subtract EQ4 from EQ5.

EQ5:      `11=6a+b`

EQ4:   `-(8=3a+b)`

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

               `3=3a`

Isolating the a, it becomes:

`3/3=(3a)/3`

`1=a`

Then, plug-in the value of a to either EQ4 or EQ5. Let's use EQ4.

`8=3a+b`

`8=3(1) + b`

`8=3+b`

`8-3=3-3+b`

`5=b`

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

`0=a+b+c`

`0=1+5+c`

`0=6+c`

`0-6=6-6+c`

`-6=c`

Now that the values of a, b and c are known, plug-in them to:

`f(n)=an^2+bn+c`

`f(n)=(1)n^2+5n+(-6)`

`f(n)=n^2+5n-6`

Replacing the f(n) with an, it becomes:

`a_n=n^2+5n-6`

Therefore, the quadratic model of the sequence is `a_n=n^2+5n-6` .