Warm-Up Find the sum of the infinite geometric series, if it exists. 𝑛=1 ∞ 8 1 5 𝑛−1 Write the repeating decimal as a fraction in lowest terms. 0.454545…

Homework Questions

12.5 Recursive Rules with Sequences Essential Question: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

Vocabulary An explicit rule gives 𝑎 𝑛 as a function of the term’s position number n in the sequence. A recursive rule gives the beginning term or terms of a sequence an then a recursive equation that tells how 𝑎 𝑛 is related to one or more preceding terms. EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

Example 1 Evaluate recursive rules Write the first six terms of the sequence and identify if it is arithmetic or geometric. 𝑎 0 =1, 𝑎 𝑛 = 𝑎 𝑛−1 +4 b. 𝑎 0 =1, 𝑎 𝑛 = 3 𝑎 𝑛−1 𝑎 0 = 𝑎 0 = 𝑎 1 = 𝑎 1 = 𝑎 2 = 𝑎 2 = 𝑎 3 = 𝑎 3 = 𝑎 4 = 𝑎 4 = 𝑎 5 = 𝑎 5 = _____________________ _____________________ EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

Example 2 Write recursive rules Write a recursive rule for the sequence. a. 3, 13, 23, 33, 43, … b. 16, 40, 100, 250, 625, … EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

c. 11, 22, 33, 44, 55, … d. 19, 13, 7, 1, -5, … EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

Example 3 Recursive rules for special sequences a. 1, 1, 2, 3, 5, … b. 1, 1, 2, 6, 24, … EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

Example 4 Applications An online music service initially has 50,000 annual members. Each year it loses 20% of its current members and adds 5,000 new members. Write a recursive rule for the number 𝑎 𝑛 of members at the start of the nth year. EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

Example 4 Applications An online music service initially has 50,000 annual members. Each year it loses 20% of its current members and adds 5,000 new members. Find the number of members at the start of the 5th year. EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.

Homework p. 830-831 # 3-21 odd EQ: How do you write a recursive rule for a sequence? Mastery demonstrated in writing in summary of notes and in practice problems.