1. 9.1: Introduction to Sequences
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9.1: Introduction to Sequences

2. Introduction to Sequences
Section 9.1
Precalculus PreAP/Dual Revised ©2017
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9.1: Introduction to Sequences

3. 9.1: Introduction to Sequences
The Factorial
Factorial is an integer greater than or equal to zero in which uses the equation, 𝒏 𝒏−𝟏 !
Represented by the symbol, “!”
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9.1: Introduction to Sequences

4. 9.1: Introduction to Sequences
Definitions
Sequence is an ordered list of numbers. (Known as the Fibonacci sequence)
Finite sequence has a limited number of terms, such as {1, 2, 3, 4}.
Infinite which continues without end such as {1, 2, 3, 4, …}
Series is where a sequence is added. A series can be finite or infinite
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9.1: Introduction to Sequences

5. 9.1: Introduction to Sequences
Example 1
Evaluate 𝟗!
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9.1: Introduction to Sequences

6. 9.1: Introduction to Sequences
Example 2
Evaluate 𝟑!𝟕! 𝟒!
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9.1: Introduction to Sequences

7. 9.1: Introduction to Sequences
Your Turn
Evaluate 𝟏𝟐! 𝟏𝟎!
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9.1: Introduction to Sequences

8. Understanding Sequences
This is a sequence having the rule, 𝒂𝒏=𝟑𝒏,
The domain gives the relative position of each term. 𝒏 𝒂𝒏
𝟏 𝟐 𝟑 𝟒 𝟓
DOMAIN:
The range gives the terms of the sequence.
𝟑 𝟔 𝟗 𝟏𝟐 𝟏𝟓
RANGE:
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9.1: Introduction to Sequences

9. 9.1: Introduction to Sequences
Example 3
Write the first six terms of the sequence of 𝒂 𝒏 =𝟐𝒏+𝟑.
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9.1: Introduction to Sequences

10. 9.1: Introduction to Sequences
Example 4
Write the first six terms of the sequence of 𝒂 𝒏 = −𝟏 𝒏+𝟏 𝟐 𝒏
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9.1: Introduction to Sequences

11. 9.1: Introduction to Sequences
Example 5
Write the first five terms of the sequence of 𝒂 𝒏 = 𝒆 𝒏 𝒏
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9.1: Introduction to Sequences

12. 9.1: Introduction to Sequences
Your Turn
Write the first six terms of the sequence of 𝒂 𝒏 = 𝒏−𝟏 𝒏
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9.1: Introduction to Sequences

13. 9.1: Introduction to Sequences
Definition
Factorial is a symbol used to multiply the previous highest number to the lowest number consecutively.
Recursive formula is a rule where the sequence is assigned a value to the first few terms and specified the nth term
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9.1: Introduction to Sequences

14. 9.1: Introduction to Sequences
Example 6
A sequence is defined recursively. Determine the first five terms of 𝒂 𝟏 =𝟑, 𝒂 𝒏 =𝟔+ 𝒂 𝒏−𝟏
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9.1: Introduction to Sequences

15. 9.1: Introduction to Sequences
Example 7
A sequence is defined recursively. Determine the first five terms of 𝒂 𝟏 =𝟓, 𝒂 𝒏 =𝟒 𝒂 𝒏−𝟏
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9.1: Introduction to Sequences

16. 9.1: Introduction to Sequences
Your Turn
A sequence is defined recursively. Determine the first five terms of 𝒂 𝟏 =−𝟔, 𝒂 𝒏 =𝒏+ 𝒂 𝒏−𝟏
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9.1: Introduction to Sequences

17. 9.1: Introduction to Sequences
Example 8
Determine the equation of this sequence: 𝟏, 𝟏 𝟓 , 𝟏 𝟐𝟓 , 𝟏 𝟏𝟐𝟓 ,…
𝒏 term
Term 𝟏
𝟏 𝟏 𝟐
𝟏 𝟓 𝟑
𝟏 𝟐𝟓 𝟒
𝟏 𝟏𝟐𝟓 𝒏
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9.1: Introduction to Sequences

18. 9.1: Introduction to Sequences
Example 9
Determine the equation of this sequence: 𝟏𝟎, −𝟐𝟎,𝟑𝟎,−𝟒𝟎,…
𝒏 term
Term 𝟏 𝟏𝟎 𝟐
−𝟐𝟎 𝟑 𝟑𝟎 𝟒
−𝟒𝟎
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9.1: Introduction to Sequences

19. 9.1: Introduction to Sequences
Example 10
Determine the equation of this sequence: 𝟏 𝟐𝟎 , 𝟐 𝟐𝟏 , 𝟑 𝟐𝟐 , 𝟒 𝟐𝟑 ,…
𝒏 term
Term 𝟏
𝟏 𝟐𝟎 𝟐
𝟐 𝟐𝟏 𝟑
𝟑 𝟐𝟐 𝟒
𝟒 𝟐𝟑 𝒏
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9.1: Introduction to Sequences

20. 9.1: Introduction to Sequences
Sigma Notation
upper limit of summation
Is read as
“the sum from k equals 1 to 5 of 3k.” 5
k = 1 ∑ 3k
= ∑ 3k 5
k = 1
index of summation
lower limit of summation
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9.1: Introduction to Sequences

21. 9.1: Introduction to Sequences
Example 11
Write out the term, 𝒌=𝟏 𝒏 𝒌!
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9.1: Introduction to Sequences

22. 9.1: Introduction to Sequences
Example 12
Express the sum of 𝟏 𝟐 + 𝟐 𝟐 + 𝟑 𝟐 + …𝟗 𝟐
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9.1: Introduction to Sequences

23. 9.1: Introduction to Sequences
Your Turn
Write out the term, 𝒌=𝟏 𝟏𝟎 𝟏 𝒌
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9.1: Introduction to Sequences

24. 9.1: Introduction to Sequences
Assignment
Page EOO, odd, odd, odd
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9.1: Introduction to Sequences