1. Arithmetic Sequences and Series
Section 9.2
Precalculus PreAP/Dual, Revised ©2016
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9.2: Arithmetic Sequences

2. 9.2: Arithmetic Sequences
Definitions
Arithmetic Sequence is a sequence whose consecutive terms have a common difference
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9.2: Arithmetic Sequences

3. 9.2: Arithmetic Sequences
Is it arithmetic?
YES NO
−𝟒𝟓, −𝟑𝟎, −𝟏𝟓, 𝟎, 𝟏𝟓…
𝟏, 𝟑, 𝟔, 𝟏𝟐, 𝟏𝟓….
𝟗 𝟒 , 𝟐, 𝟕 𝟒 , 𝟑 𝟐 , 𝟓 𝟒 …
𝟏 𝟑 , 𝟐 𝟑 , 𝟒 𝟑 , 𝟖 𝟑 , 𝟏𝟔 𝟑 …
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9.2: Arithmetic Sequences

4. 9.2: Arithmetic Sequences
Steps
Plug the equation for the nth term of an arithmetic sequence: 𝒂 𝒏 = 𝒂 𝟏 +𝒅 𝒏−𝟏
𝒂 𝟏 is the first sequence number
𝒅 is the common difference
To determine the common difference, ensure that all of the values have a common difference
Simplify the equation
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9.2: Arithmetic Sequences

5. 9.2: Arithmetic Sequences
Example 1
Determine the arithmetic sequence equation of 𝟕, 𝟏𝟏, 𝟏𝟓, 𝟏𝟗…
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9.2: Arithmetic Sequences

6. 9.2: Arithmetic Sequences
Example 1
Determine the arithmetic sequence equation of 𝟕, 𝟏𝟏, 𝟏𝟓, 𝟏𝟗…
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9.2: Arithmetic Sequences

7. 9.2: Arithmetic Sequences
Example 2
The arithmetic sequence, 𝒂 𝒏 =𝟓−𝟑𝒏 is given. What is the common difference and the first 4 terms?
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9.2: Arithmetic Sequences

8. 9.2: Arithmetic Sequences
Your Turn
Determine the arithmetic sequence equation of 𝟏𝟗, 𝟏𝟐, 𝟓,−𝟐, −𝟗, …
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9.2: Arithmetic Sequences

9. 9.2: Arithmetic Sequences
Example 3
Determine the equation whose 𝒏th term of the arithmetic where the first term is 𝟐 and common difference is 𝟑.
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9.2: Arithmetic Sequences

10. 9.2: Arithmetic Sequences
Example 4
Determine the equation whose 𝒏th term of the arithmetic where the first term is 𝟏𝟕 and common difference is 𝟏𝟕 .
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9.2: Arithmetic Sequences

11. 9.2: Arithmetic Sequences
Your Turn
Determine the equation whose 𝒏th term of the arithmetic where the first term is −𝟏𝟎 and common difference is −𝟓𝟎.
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9.2: Arithmetic Sequences

12. 9.2: Arithmetic Sequences
Example 5
Determine its first five terms from the given equation, 𝒂 𝒏 =𝟏𝟐+𝟕𝒏. Then, determine the 20th term.
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9.2: Arithmetic Sequences

13. 9.2: Arithmetic Sequences
Example 6
If the common difference is 𝟒 and the fifth term is 𝟏𝟓, what is the 10th term of an arithmetic sequence? Do we know what the first term is?
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9.2: Arithmetic Sequences

14. 9.2: Arithmetic Sequences
Example 6
If the common difference is 𝟒 and the fifth term is 𝟏𝟓, what is the 10th term of an arithmetic sequence?
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9.2: Arithmetic Sequences

15. 9.2: Arithmetic Sequences
Your Turn
If the common difference is 𝟗 and the 𝟓𝒕𝒉 term is 𝟕𝟑, what is the 10th term of an arithmetic sequence?
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9.2: Arithmetic Sequences

16. upper limit of summation 9.2: Arithmetic Sequences
Sigma Notation
upper limit of summation
Is read as
“the sum from 𝒏 equals 𝟏 to 𝟓 of 𝟑𝒏.” 𝟓
𝒌=𝟏 ∑ 3𝑛
= ∑ 3n 5
k = 1
index of summation
lower limit of summation
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9.2: Arithmetic Sequences

17. Steps for Arithmetic Series
Identify the number of terms, lower and upper limit of the summation
Then, identify the first and last term by plugging the given equation: 𝒌=𝟏 𝒏 𝒂 𝒌 =𝒏 𝒂 𝟏 + 𝒂 𝒏 𝟐
𝒂 𝒌 is the given equation
𝒂 𝟏 is the first term of the sequence
𝒂 𝒏 is the last term of the sequence
𝒏 = term
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9.2: Arithmetic Sequences

18. 9.2: Arithmetic Sequences
Example 7
Find the following sum, 𝟏+𝟑+𝟓+𝟕+𝟗+𝟏𝟏+𝟏𝟑+𝟏𝟓+𝟏𝟕+𝟏𝟗+𝟐𝟏
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9.2: Arithmetic Sequences

19. 9.2: Arithmetic Sequences
Example 8
Find the sum of the first 100 terms of the arithmetic sequence, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, …
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9.2: Arithmetic Sequences

20. 9.2: Arithmetic Sequences
Your Turn
Find the following sum of the total of 14 terms, 𝟏+𝟓+𝟗+…+𝟒𝟗+𝟓𝟑
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9.2: Arithmetic Sequences

21. 9.2: Arithmetic Sequences
Example 9
Evaluate 𝒌=𝟏 𝟓 𝟑𝒌
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9.2: Arithmetic Sequences

22. 9.2: Arithmetic Sequences
Example 9
Evaluate 𝒌=𝟏 𝟓 𝟑𝒌
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9.2: Arithmetic Sequences

23. 9.2: Arithmetic Sequences
Example 10
Evaluate 𝒌=𝟏 𝟏𝟐𝟎 𝟓− 𝟏 𝟒 𝒌
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9.2: Arithmetic Sequences

24. 9.2: Arithmetic Sequences
Your Turn
Evaluate 𝒌=𝟐 𝟔 𝟐𝒌+𝟏
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9.2: Arithmetic Sequences

25. 9.2: Arithmetic Sequences
Example 11
Determine the amount of terms in this arithmetic series, 𝒂 𝟏 =𝟏𝟗, 𝒂 𝒏 =𝟗𝟔, 𝑺 𝒏 𝑺𝒖𝒎 =𝟔𝟗𝟎
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9.2: Arithmetic Sequences

26. 9.2: Arithmetic Sequences
Your Turn
Determine the amount of terms in this arithmetic series, 𝒂 𝟏 =𝟕, 𝒂 𝒏 = 𝟐𝟓, 𝑺𝒏 𝑺𝒖𝒎 =𝟏𝟎
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9.2: Arithmetic Sequences

27. 9.2: Arithmetic Sequences
Assignment
Page EOO, odd
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9.2: Arithmetic Sequences