1. 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, , , , …..
−4, −8, −12, −16, ….
G, 𝑡 𝑛 = 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

2. 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

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

4. 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.)

5. 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.

6. 13.2 Recursive Definitions
To use sequences defined recursively
to solve problems.
Find the first 5 terms.
Initial Condition
Recursion Equation

7. 13.2 Recursive Definitions
Arithmetic
Recursive Definition?
Explicit Definition?

8. 13.2 Recursive Definitions
Give the first 5 terms of the sequence.
What kind of sequence is it?
Arithmetic
Find an explicit formula.

9. 13.2 Recursive Definitions
Give the first 5 terms of the sequence.
What kind of sequence is it?
Geometric
Find an explicit formula.

10. 13.2 Recursive Definitions

11. 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−

12. 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. 13.2 Recursive Definitions
Arithmetic?
Geometric?

14. Classwork
Give the first 4 terms of each sequence. 1. 𝑡 1 =5; 𝑡 𝑛 =𝑡 𝑛− 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 , 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

15. Homework
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