Section 8.5 Using Recursive Rules and Sequences Honors Algebra 2 Section 8.5 Using Recursive Rules and Sequences

Has anyone heard of the Fibonacci sequence?

This applies to bunny reproduction hence the saying

Fibonacci Sequence in nature The center of the sunflower replicates the Fibonacci Sequence.

Recursive rule-gives the first term of a sequence and a recursive equation that tells how 𝑎 𝑛 is related to one or more previous terms

Working with a partner, do exploration 1 on page 441.

Write a recursive rule- #1 Name the first term #2 Give a formula to use a previous term to find a new term Working with a partner, do Exploration 2 on page 441

Find the first six terms of the following sequence 𝑎 0 =81 𝑎 𝑛 = 1 3 𝑎 𝑛−1 81 is the first term 𝑎 𝑛−1 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚 𝑏𝑒𝑓𝑜𝑟𝑒 𝑎 𝑛 You always have to find the terms in order!

Write the first six terms of the sequence 𝑓 1 =23 𝑓 𝑛 =𝑓 𝑛−1 −6 This is a recursive rule!!!

Explicit rule-gives 𝑎 𝑛 as a function of the term’s position number in the sequence. 𝑎 𝑛 =5+2𝑛 𝑎 1 =5+2 1 =7 𝑎 2 =5+2 2 =9 𝑎 3 =5+2 3 =11 We can find any number term using an explicit rule.

Write a recursive rule for the sequence with the following explicit rule #1 𝑎 𝑛 =30−5𝑛 #2 𝑎 𝑛 =7 (2) 𝑛−1

Write an explicit rule for a sequence with the following recursive rule #1 𝑎 1 =7, 𝑎 𝑛 = 𝑎 𝑛−1 +4 #2 𝑎 1 =5, 𝑎 𝑛 =6 𝑎 𝑛−1

Real-life A mosquito population in a controlled laboratory condition is estimated to be about 500. Each day an additional 100 mosquitoes are hatched. The population also declines by 85% every day from a pesticide and other natural causes. Write a recursive rule for the number 𝑎 𝑛 of mosquitoes at the start of the nth day

Find the number of mosquitoes after the fifth day.

Describe what happens to the number of mosquitoes over time Describe what happens to the number of mosquitoes over time. (A calculator helps here)

Method #1 Press MODE, on the 4th line arrow to SEQ, on the 5th line arrow to DOT, press ENTER Press y= For nmin press 0 For u(n) type .15u(n-1)+100 (the X,T,𝜃,n) will type n and get u by pressing 2nd 7 For u(nMin) type {500, .15(500)+100} This is the first two terms Press table or graph (check window)

Method #2 Type 500, Press ENTER Type .15 2nd (-) +100 Each time you press ENTER, you get the next number in the sequence.

You borrow $20,000 at 8% annual interest compounded monthly for 4 years. The monthly payment is $489. Write a recursive rule. What is 𝑎 1 ? What is the monthly interest rate (rounded to four decimal places)?

Find the balance after the fourth payment. Find the amount of the last payment.

Assignment #38 Pg. 447 #3-39 multiples of 3, 40, 42,45,48, 51, 52, 53, 57