1. RECURSIVE FORMULAS

2. 43210 In 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 (n th ) in the sequence. With help from the teacher, the student has partial 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.

3. 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 a1a1 2 a2a2 4 a3a3 6 a4a4 8 a n = 2n

4. 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 (a 1 = 2) we can display this in a recursive formula using the following: a n = a n-1 + 2 a n = term number and a n-1 = the term before the n term

5. HOW DOES A RECURSIVE FORMULA WORK? a n = a n-1 + 2 2, 4, 6, 8… The 4 th term in this sequence 8. (a 4 = 8) Find the 5 th term. a 5 = a (5-1) + 2 a 5 = a 4 + 2 a 5 = 8 + 2 a 5 = 10 a n = a n-1 + 2 The 5 th term in this sequence 10. (a 5 = 10) Find the 6 th term. a 6 = a (6-1) + 2 a 6 = a 5 + 2 a 6 = 10 + 2 a 6 = 12

6. USE THE RECURSIVE FORMULA TO WRITE THE 1 ST FIVE TERMS OF THE SEQUENCE. a n = a n-1 – 2, a 1 = 27 We are provided the 1 st term of the sequence, 27. We need to find the next four terms. a 2 = 27 – 2 a 2 = 25 a 3 = 25 – 2 a 3 = 23 a 4 = 23 – 2 a 4 = 21 a 5 = 21 – 2 a 5 = 19 The first five terms of the sequence are 27, 25, 23, 21, and 19.

7. 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? a n = a (n-1) + a (n-2) a 10 = a 9 + a 8 a 10 = 34 + 21 a 10 = 55 Sequence Term Term a1a1 1 a2a2 1 a3a3 2 a4a4 3 a5a5 5 a6a6 8 a7a7 13 a8a8 21 a9a9 34

8. 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. a n = a (n-1) + 2, where a 1 = 12 Why do we have to say what a 1 is? a n = 2n + 10

9. 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? a n = -8n + 17 Sequence Term Term a1a1 9 a2a2 1 a3a3 -7 a4a4 -15

10. 9, 1, -7, -15… Write a recursive formula for the sequence. You subtract 8 to get to the next term. a n = a (n-1) - 8 Which formula would you use to find the 38 th term? The explicit formula is best for finding specific terms in a sequence. a n = -8n + 17 a n = -8n + 17 a 38 = -8(38) + 17 a 38 = -304 + 17 a 38 = - 287