RECURSIVE FORMULAS

4 3 2 1 0In addition to level 3.0 and above and beyond what was taught in class,  the student may:· Make connection with other concepts in math· Make connection with other content areas.

The student will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences. - Linear and exponential functions can be constructed based off a graph, a description of a relationship and an input/output table. - Write explicit rule for a sequence. - Write recursive rule for a sequence.

The student will be able to:- Determine if a sequence is arithmetic or geometric. - Use explicit rules to find a specified term (nth) in the sequence.  

With help from theteacher, the student haspartial success with building a function that models a relationship between two quantities.

Even with help, the student has no success understanding building functions to model relationship between two quantities.

Focus 7 Learning Goal – (HS.F-BF.A.1, HS.F-BF.A.2, HS.F-LE.A.2, HS.F-IF.A.3) = Students will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences.

EXPLICIT FORMULA (REVIEW)

An explicit formula allows you to determine any term in a set sequence.

Write the explicit formula for the sequence: 2, 4, 6, 8…

What is the pattern? How is each term related to the term number?

The explicit formula is:

Sequence Term

Term

a1 2

a2 4

a3 6

a4 8an = 2n

RECURSIVE FORMULA A recursive formula always uses the preceding term to define the next term of the sequence.

Write the recursive formula for the sequence: 2, 4, 6, 8…

A recursive formula tells us how each term is connected to the next term.

The difference between each term is 2 (a1 = 2) we can display this in a recursive formula using the following:

an = an-1 + 2

an = term number and an-1 = the term before the n term

HOW DOES A RECURSIVE FORMULA WORK? an = an-1 + 2

2, 4, 6, 8… The 4th term in this sequence 8. (a4 = 8)

Find the 5th term.

a5 = a(5-1) + 2

a5 = a4 + 2

a5 = 8 + 2

a5 = 10

an = an-1 + 2 The 5th term in this sequence 10. (a5 = 10)

Find the 6th term.

a6 = a(6-1) + 2

a6 = a5 + 2

a6 = 10 + 2

a6 = 12

USE THE RECURSIVE FORMULA TO WRITE THE 1ST FIVE TERMS OF THE SEQUENCE. an = an-1 – 2, a1 = 27

We are provided the 1st term of the sequence, 27. We need to find the next four terms.

a2 = 27 – 2

a2 = 25

a3 = 25 – 2

a3 = 23

a4 = 23 – 2

a4 = 21

a5 = 21 – 2

a5 = 19

The first five terms of the sequence are 27, 25, 23, 21, and 19.

1, 1, ,2, 3, 5, 8, 13, 21, 34, …

One of the most famous sequences is the Fibonacci sequence.

How is each term generated?

What would be the next term?

an = a(n-1) + a(n-2)

a10 = a9 + a8

a10 = 34 + 21

a10 = 55

Sequence Term

Term

a1 1

a2 1

a3 2

a4 3

a5 5

a6 8

a7 13

a8 21

a9 34

Write the first 5 terms of the sequence using the explicit formula given. Then, write the recursive formula for the sequence.

Substitute the term numbers 1 through 5 for “n” to write the first 5 terms of the sequence.

12, 14, 16, 18, 20

How would you write the recursive formula?

Each term is increased by 2. Just add two to the previous term.

an = a(n-1) + 2, where a1 = 12

Why do we have to say what a1 is?

an = 2n + 10

9, 1, -7, -15… Write an explicit formula for the sequence. Since the sequence is subtract 8, you need to multiply the term number by -8. What do you need to do next in order to get to the first number?

an = -8n + 17

Sequence Term

Term

a1 9

a2 1

a3 -7

a4 -15

9, 1, -7, -15… Write a recursive formula for the sequence.

You subtract 8 to get to the next term.

an = a(n-1) - 8 Which formula would you use to find the 38th term?

The explicit formula is best for finding specific terms in a sequence. an = -8n + 17

an = -8n + 17

a38 = -8(38) + 17

a38 = -304 + 17

a38 = - 287