Is the sequence arithmetic, geometric, or neither Is the sequence arithmetic, geometric, or neither? Find a formula for 𝑡 𝑛 and evaluate 𝑡 8 . Warm Up N, 𝑡 𝑛 =𝑛 𝑛+1 , 𝑡 8 =72 1∙2, 2∙3, 3∙4, 4∙5,…. 3 5, 3 10, 3 20, 3 40, ….. −4, −8, −12, −16, …. G, 𝑡 𝑛 = 3 5 3 2 𝑛−1 𝑡 8 =4 3 10 A, 𝑡 𝑛 =−4𝑛, 𝑡 8 =−32 In a certain sequence, 𝑡 2 =2 and 𝑡 5 =16. 118 3 4. Find 𝑡 10 if the sequence is arithmetic. 5. Find 𝑡 10 if the sequence is geometric. 512

In a certain sequence, 𝑡 2 =2 and 𝑡 5 =16. 4. Find 𝑡 10 if the sequence is arithmetic. d= 16−2 5−2 = 14 3

Write and solve a system of equations. In a certain sequence, 𝑡 2 =2 and 𝑡 5 =16. 5. Find 𝑡 10 if the sequence is geometric. Write and solve a system of equations. 𝑡𝑛 = 𝑡1∙ 𝑟 𝑛−1 Divide the equations: 8= 𝑟 3 16=𝑡1⋅ 𝑟 4 2=𝑡1⋅𝑟 𝑟=2 𝑡𝑛=1⋅ 2 𝑛−1 2=𝑡1⋅2 𝑡10=1⋅ 2 10−1 =512 𝑡1=1

13.2 Recursive Definitions To use sequences defined recursively to solve problems. Recursion is the process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term. A recursive formula always has two parts:   1.  the starting value for t1.   2.  the recursion equation for tn as a function of tn-1  (the term before it.)

Consider the sequence  2, 4, 6, 8, 10, ... Explicit formula: Recursive formula: 𝑡 1 =2 𝑡 𝑛 =2𝑛 𝑡 𝑛 = 𝑡 𝑛−1 +2 Certain sequences, such as this arithmetic sequence, can be represented in more than one manner.  This sequence can be represented as either an explicit (general) formula or a recursive formula.

13.2 Recursive Definitions To use sequences defined recursively to solve problems. Find the first 5 terms. Initial Condition Recursion Equation

13.2 Recursive Definitions Arithmetic Recursive Definition? Explicit Definition?

13.2 Recursive Definitions Give the first 5 terms of the sequence. What kind of sequence is it? Arithmetic Find an explicit formula.

13.2 Recursive Definitions Give the first 5 terms of the sequence. What kind of sequence is it? Geometric Find an explicit formula.

13.2 Recursive Definitions

Consider the sequence 2, 5, 26, 677, ... 𝑡 1 =2 𝑡 𝑛 = 𝑡 𝑛−1 2 +1 This sequence is neither arithmetic nor geometric.  It does, however, have a pattern of development based upon each previous term. Recursive formula: 𝑡 1 =2 𝑡 𝑛 = 𝑡 𝑛−1 2 +1

13.2 Recursive Definitions The annual population growth is 1% of those already in the country. This growth rate equals the birth rate minus the death rate. 20,000 people immigrate into the country each year.

13.2 Recursive Definitions Arithmetic? Geometric?

Classwork Give the first 4 terms of each sequence. 1. 𝑡 1 =5; 𝑡 𝑛 =𝑡 𝑛−1 +3 2. 𝑡 1 =10; 𝑡 𝑛 =𝑡 𝑛−1 +𝑛 3. 𝑡 1 =4; 𝑡 𝑛 = 𝑡 𝑛−1 2 −10 Give a recursive definition for the sequence. 4. 81, 27, 9, 3, …. 5. 1, 2, 6, 24, 120, …… 5, 8, 11, 14 10, 12, 15, 19 26, 666, 443, 546 𝑡 1 =81; 𝑡 𝑛 = 1 3 𝑡 𝑛−1 𝑡 1 =1; 𝑡 𝑛 =𝑛∙𝑡 𝑛−1

Homework Page 481 #1-29 odds