1. 7-8 Notes for Algebra 1
Recursive Formulas

2. 7-8 pg

3. Recursive Formula
Allows you to find the nth term of a sequence by performing operations to one or more of the preceding terms.

4. Example 1: Use a Recursive Formula
Find the first five terms of the sequence in which 𝑎 1 =−8 and 𝑎 𝑛 =−2 𝑎 𝑛−1 +5, if 𝑛≥2.

5. Example 1: Use a Recursive Formula
Find the first five terms of the sequence in which 𝑎 1 =−8 and 𝑎 𝑛 =−2 𝑎 𝑛−1 +5, if 𝑛≥2. 𝑎 2 =21 𝑎 2 =−2 −8 +5 𝑎 2 =−37 𝑎 2 =− 𝑎 2 =79 𝑎 2 =−2 −37 +5 𝑎 2 =−153 𝑎 2 =−

6. Writing Recursive Formulas
Determine if the sequence is arithmetic or geometric by finding a common difference or common ratio.
Write a recursive formula Arithmetic Sequence 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 Geometric Sequence 𝑎 𝑛 = 𝑟∙𝑎 𝑛−1
State the first term and domain for 𝑛

7. Example 2: Write Recursive Formulas
Write a recursive formula for each sequence. 1.) 23, 29, 35, 41, … 2.) 7, −21, 63, −189, …

8. Example 2: Write Recursive Formulas
Write a recursive formula for each sequence. 1.) 23, 29, 35, 41, … 𝑎 1 =23; 𝑎 𝑛 = 𝑎 𝑛−1 +6; 𝑛≥2 2.) 7, −21, 63, −189, … 𝑎 1 =7; 𝑎 𝑛 = −3∙𝑎 𝑛−1 ; 𝑛≥2

9. Example 3: Write Recursive and Explicit Formulas
CARS The price of a car depreciates at the end of each year. a.) Write a recursive formula for the sequence b.) Write an explicit formula for the sequence
Year
Price ($) 1
12,000 2
7200 3
4320 4
2592

10. Example 3: Write Recursive and Explicit Formulas
CARS The price of a car depreciates at the end of each year. a.) Write a recursive formula for the sequence 𝑎 1 =12,000; 𝑎 𝑛 =0.6 𝑎 𝑛−1 b.) Write an explicit formula for the sequence 𝑎 𝑛 =12, 𝑛−1
Year
Price ($) 1
12,000 2
7200 3
4320 4
2592

11. Example 4: Translate between Recursive and Explicit Formulas
1.) Write a recursive formula for 𝑎 𝑛 =2𝑛−4. 2.) Write an explicit formula for 𝑎 1 =84; 𝑎 𝑛 =1.5 𝑎 𝑛−1 ; 𝑛≥2

12. Example 4: Translate between Recursive and Explicit Formulas
1.) Write a recursive formula for 𝑎 𝑛 =2𝑛−4. 𝑎 1 =−2; 𝑎 𝑛 = 𝑎 𝑛−1 +2; 𝑛≥2 2.) Write an explicit formula for 𝑎 1 =84; 𝑎 𝑛 =1.5 𝑎 𝑛−1 ; 𝑛≥2 𝑎 𝑛 = 𝑛−1 ; 𝑛≥2