Recursive and explicit equations for a quadratic sequence

Question

I cannot figure out how to form equations for a quadratic sequence. Not explicit nor recursive. The sequence reads 5,10,17

Answer 1

Hi Ash,

 

The standard form of a quadratic function is:

    f(x) = ax² + bx + c

 

We are given three values for our quadratic function - the first three terms of our sequence:

    f(1) = 5

    f(2) = 10

    f(3) = 17

 

This gives us three equations to solve for our three unknowns (coefficients a, b, and c):

    a(1)² + b(1) + c = 5 (1st term)

    a(2)² + b(2) + c = 10 (2nd term)

    a(3)² + b(3) + c = 17 (3rd term)

 

After evaluating our three plug-ins for x (sequence number of each term in our sequence), we get our system of three linear equations to solve for our three unknowns (a, b, and c):

    Equation #1:  a + b + c = 5

    Equation #2:  4a + 2b + c = 10

    Equation #3:  9a + 3b + c = 17

 

Subtracting Equation #1 from Equation #2 to eliminate c, we get Equation #4:

    Equation #4:  3a + b = 5

 

Subtracting Equation #2 from Equation #3 to again eliminate c, we get Equation #5:

    Equation $5:  5a + b = 7

 

Subtracting Equation #4 from Equation #5 to eliminate b, we get Equation #6 which allows us to solve for a:

    Equation #6:  2a = 2

                          a = 2/2

                          a = 1

 

Plugging our value for a into Equation #5 allows us to solve for b:

    Equation #5:  5a + b = 7

                        5(1) + b = 7

                        5 + b = 7

                              b = 7 - 5

                              b = 2

 

Plugging our values for a and b into Equation #1 allows us to solve for c:

    Equation #1:  a + b + c = 5

                        1 + 2 + c = 5

                        3 + c = 5

                              c = 5 - 3

                              c = 2

 

Having all our values for coefficients a, b, and c we can write our quadratic sequence formula, using n (sequence number) in place of our x variable:

    ax² + bx + c (standard quadratic function)

    a = 1; b = 2; c = 2; n = x

    n² + 2n + 2 (quadratic sequence formula)

 

Let's test our formula on the three given sequence terms:

   1st term (n = 1):

       (1)² + 2(1) + 2 = 1 + 2 + 2 = 5 (checks out)

   2nd term (n = 2):

       (2)² + 2(2) + 2 = 4 + 4 + 2 =10 (checks out)

   3rd term (n = 3):

       (3)² + 2(3) + 2 = 9 + 6 + 2 = 17 (checks out)

 

Since our quadratic sequence formula gave the correct values for all three given terms in our sequence, we are confident that our quadratic sequence formula is correct.

 

This was a lengthy procedure for sure, but a guaranteed way to figure out any quadratic sequence formula as long as we are given at least the first three terms of the sequence.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.