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.

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

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

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

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

translating between Recursive and Explicit Rules EX: Write an explicit rule for the given recursive rule. 𝑎 1 =3, 𝑎 𝑛−1 +5 Sequence: 3, 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

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 =−2 𝑟= −3 𝑎 1 =−2, 𝑎 𝑛 = 𝑎 𝑛−1 −3