7-8 Notes for Algebra 1 Recursive Formulas

7-8 pg. 448 10-21

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

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.

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 =−2 21 +5 𝑎 2 =79 𝑎 2 =−2 −37 +5 𝑎 2 =−153 𝑎 2 =−2 79 +5

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

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

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

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

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 =12,000 0.6 𝑛−1 Year Price ($) 1 12,000 2 7200 3 4320 4 2592

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

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 =84 1.5 𝑛−1 ; 𝑛≥2