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

2. Homework Questions

3. 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.

4. 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.

5. 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.

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

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. Homework p # 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.