12.5 use recursive rules w/ sequences & functions

Explicit/recursive rules Explicit Rule: a rule that gives 𝑎 𝑛 as a function of the term’s position number n in the sequence. Recursive Rule: a rule that gives the beginning term or terms of a sequence and then a recursive equation that tells how 𝑎 𝑛 is related to one or more preceding terms.

Evaluate recursive rules 1. Write the first six terms of the sequence. a) 𝑎 0 =1, 𝑎 𝑛 = 𝑎 𝑛−1 +4 b) 𝑎 1 =32, 𝑎 𝑛 = 1 2 𝑎 𝑛−1

Recursive equations

Writing recursive rules 2. Write a recursive rule for the sequence. a) 100, 40, 16, 32 5 , 64 25 , … b) 8, 28, 48, 68, 88, …

Writing recursive rules for special sequences 3. Write a recursive rule for the sequence. a) -10, 8, 18, 10, -8, -18 b) 1, 1 2 , 1 6 , 1 24 , 1 120 ,…

Solving a word problem 4. 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 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 5th day. Describe what happens to the number of mosquitoes over time.

iteration Iteration: involves the repeated composition of a function f with itself. Example: 5. Find the first three iterates 𝑥 1 , 𝑥 2, and 𝑥 3 of the function 𝑓 𝑥 =5𝑥−3 for an initial value of 𝑥 0 =1.