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Pre Cal 103 Lesson with notes 5th.notebook
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May 09, 2017
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103 Geometric Sequences and Series
Daily Outcomes:I can find nth terms and geometric means of geometric
sequencesI can find sums of n terms of geometric series and the sums of
infinite geometric series
What is the relationship in revenue from one year to the next?
How can the amount of next year's revenue be predicted?
If the revenue in 1997 was $500,000, what was the revenue in 2007 if the pattern continued?
Apr 84:44 PM
Geometric sequence: sequence in which the ratio between successive terms is a constant
This constant is called the common ratio, r
8, 2, 1/2, . . .
Pre Cal 103 Lesson with notes 5th.notebook
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Example 1: Determine the common ratio, and find the next three terms of each geometric sequence.
1. 6, 9, 13.5, . . . 2. 243n 729, 81n + 243, 27n 81, . . .
3. 4, 11, 30.25, . . . 4. 64r 128, 16r + 32, 4r 8, . . .
Pre Cal 103 Lesson with notes 5th.notebook
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Recursively: where a term an is found by taking the product of the previous term an1 and r, or an = an1r
Explicit formula: consider the pattern created by the geometric sequence for which a1 = 3 and r = 4.
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Pre Cal 103 Lesson with notes 5th.notebook
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To write the explicit formula: use the formula and fill in a1 and the common ratio.
To write a recursive formula: state the first term (a1 = ?), then write the formula as the common ratio times the previous term (an1)
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Example 2: Write an explicit formula and a recursive formula for finding the nth term in the sequence.
5. 1, 2, 4, . . . 6. 2, 25, 312.5, . . .
Pre Cal 103 Lesson with notes 5th.notebook
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Example 3: Find the nth term of a geometric sequence by using an explicit formula
7. Find the 11th term of the geometric sequence 122, 115.9, 110.105, . . .
Find the specified term of each geometric sequence or sequence with the given characteristics.
8. a9 for 4, 14, 49, . . . 9. a12 if a3 = 32 and r = 4
Pre Cal 103 Lesson with notes 5th.notebook
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Example 4:
10. A couple purchased a home for $225,000. At the end of each year, the value of the home appreciates 3%.
a. Write an explicit formula for finding the value of the home after n years.
b. What is the value of the home after the tenth year?
Pre Cal 103 Lesson with notes 5th.notebook
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11. Rohan purchased a personal watercraft for $9000. Assume that by the end of each year, the value of the watercraft depreciates 30%.
a. Write an explicit formula for finding the value of Rohan's watercraft after n years.
b. What is the value of Rohan's watercraft after 5 years?
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Geometric means: two nonconsecutive terms of a geometric sequence
Pre Cal 103 Lesson with notes 5th.notebook
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Example 5: Find the indicated geometric means for each pair of nonconsecutive terms.
12. 264 and 1.03125, 3 means 13. 4 and 13.5; 2 means
14. 10 and 0.016; 3 means
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Geometric series: a sum of the terms of a geometric sequence
Pre Cal 103 Lesson with notes 5th.notebook
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Pre Cal 103 Lesson with notes 5th.notebook
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Example 6:
15. Find the sum of the first eleven terms of the geometric series 4, 6, 9, . . .
16. Find the sum of the first n terms of a geometric series with a1 = 4, an = 65,536, and r = 2
17. Find the sum of the first 11 terms of the geometric series 7 + (24.5) + 85.75 + . . .
18. Find the sum of the first n terms of a geometric series with a1 = 8, an = 131,072, and r = 4
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Pre Cal 103 Lesson with notes 5th.notebook
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Example 7: Find each sum.
19. 20. 21.
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The formula for the sum of a finite geometric series can be used to develop a formula for the sum of an infinite geometric series. If |r| > 1, then |rn| increases without limit as n . However, when |r| < 1, rn approaches 0 as n .
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Pre Cal 103 Lesson with notes 5th.notebook
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Example 8: If possible, find the sum of each infinite geometric series.
22. 24 + 18 + 13.5 + . . . 23. 0.33 + 0. + 1.32 + . . .
24. 25.
26. 10 + (5) + 2.5 + . . . 27. 20 + 15 + 10 + . . .
