1. Lesson 6.7 Recursive Sequences
Topic/Objective: To write terms in a recursive sequence.
To recursive rules for sequences
To write explicit rules
Explicit rule: gives 𝑎 𝑛 as a function of the term’s position number n in a sequence. For example: in the sequence 4, 6, 8, 10… the explicit rule is 𝑎 𝑛 =2n+2
Recursive Rule: gives beginning term and a recursive equation that tells how 𝑎 𝑛 is related to preceding terms.

2. Recursive Equation for an Arithmetic Sequence
𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 where d is the common difference
EX: Write the first 5 terms of the sequence when 𝑎 1 = 4 with a recursive equation of 𝑎 𝑛 = 𝑎 𝑛−1 +3
𝑎 1 =4
𝑎 2 =7
𝑎 3 =10
𝑎 4 =13
𝑎 5 =16
Arithmetic: You can find each term if you know the previous term

3. Recursive Equation for a geometric sequence
𝑎 𝑛 =𝑟∙ 𝑎 𝑛−1 where r is the common ratio
Write the first 5 terms of the sequence where 𝑎 1 =2 and 𝑎 𝑛 =3 𝑎 𝑛−1
𝑎 1 =2
𝑎 2 =6
𝑎 3 =18
𝑎 4 =54
𝑎 5 =162

4. Write the recursive rule for a sequence
-14, -6, 2, 10, 18
𝑎 1 = Common difference is 8
𝑎 𝑛 = 𝑎 𝑛−1 +𝑑
𝑎 1 =−14, 𝑎 𝑛 = 𝑎 𝑛−1 +8

5. Write the recursive rule for a sequence
128, 64, 32, 16, 8
𝑎 1 = Common ratio is .5
𝑎 𝑛 =𝑟∙ 𝑎 𝑛−1
𝑎 1 = 128, 𝑎 𝑛 =.5∙ 𝑎 𝑛−1

6. translating between Recursive and Explicit Rules
EX: Write an explicit rule for the given recursive rule.
𝑎 1 =3, 𝑎 𝑛−1 +5
Sequence: , 8, 13, 18
Use the sequence to write explicit rule.
𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑
𝑎 𝑛 = 3 +(n – 1)5
𝑎 𝑛 = 3 + 5n – 5
𝑎 𝑛 =5𝑛−2

7. Write a recursive rule for the explicit Rule.
𝑎 𝑛 =−3𝑛+1
Sequence: -2, -5, -8, -11
Use Sequence to write recursive rule.
𝑎 1 =− 𝑟= −3
𝑎 1 =−2, 𝑎 𝑛 = 𝑎 𝑛−1 −3