Geometric Sequence and Series

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Sharp Tutor Fri, Jan 14, 2022 10:34 PM UTC

This ppt. gives an introduction to Geometric Sequence and Series

1. 9-1
9-1 Geometric
GeometricSequences
Sequences
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
Holt McDougal
Algebra 1Algebra
McDougal Algebra11

2. 9-1 Geometric Sequences
Warm Up
Find the value of each expression.
1. 25 32 2. 2–5
3. –34 –81 4. (–3)4 81
5. (0.2)3 0.008 6. 7(–4)2 112
7. 8. 12(–0.4)3 –0.768
Holt McDougal Algebra 1

3. 9-1 Geometric Sequences
Objectives
Recognize and extend geometric
sequences.
Find the nth term of a geometric
sequence.
Holt McDougal Algebra 1

4. 9-1 Geometric Sequences
Vocabulary
geometric sequence
common ratio
Holt McDougal Algebra 1

5. 9-1 Geometric Sequences
The table shows the heights of a bungee
jumper’s bounces.
The height of the bounces shown in the table
above form a geometric sequence. In a
geometric sequence, the ratio of successive
terms is the same number r, called the
common ratio.
Holt McDougal Algebra 1

6. 9-1 Geometric Sequences
Geometric sequences can be thought of as
functions. The term number, or position in the
sequence, is the input, and the term itself is the
output.
1 2 3 4 Position
3 6 12 24 Term
a1 a2 a3 a4
To find a term in a geometric sequence, multiply
the previous term by r.
Holt McDougal Algebra 1

7. 9-1 Geometric Sequences
Holt McDougal Algebra 1

8. 9-1 Geometric Sequences
Writing Math
The variable a is often used to represent terms in
a sequence. The variable a4 (read “a sub 4”)is the
fourth term in a sequence.
Holt McDougal Algebra 1

9. 9-1 Geometric Sequences
Example 1A: Extending Geometric Sequences
Find the next three terms in the geometric
sequence.
1, 4, 16, 64,…
Step 1 Find the value of r by dividing each term
by the one before it.
1 4 16 64
The value of r is 4.
Holt McDougal Algebra 1

10. 9-1 Geometric Sequences
Example 1A Continued
Find the next three terms in the geometric
sequence.
1, 4, 16, 64,…
Step 2 Multiply each term by 4 to find the next
three terms.
64 256 1024 4096
4 4 4
The next three terms are 256, 1024, and 4096.
Holt McDougal Algebra 1

11. 9-1 Geometric Sequences
Example 1B: Extending Geometric Sequences
Find the next three terms in the geometric
sequence.
Step 1 Find the value of r by dividing each term
by the one before it.
The value
– of r is .
Holt McDougal Algebra 1

12. 9-1 Geometric Sequences
Helpful Hint
When the terms in a geometric sequence
alternate between positive and negative, the
value of r is negative.
Holt McDougal Algebra 1

13. 9-1 Geometric Sequences
Example 1B Continued
Find the next three terms in the geometric
sequence.
Step 2 Multiply each term by to find the next
three terms.
The next three terms are
Holt McDougal Algebra 1

14. 9-1 Geometric Sequences
Check It Out! Example 1a
Find the next three terms in the geometric
sequence.
5, –10, 20,–40,…
Step 1 Find the value of r by dividing each term
by the one before it.
5 –10 20 –40
The value of
r is –2.
Holt McDougal Algebra 1

15. 9-1 Geometric Sequences
Check It Out! Example 1a Continued
Find the next three terms in the geometric
sequence.
5, –10, 20,–40,…
Step 2 Multiply each term by –2 to find the next
three terms.
–40 80 –160 320
(–2) (–2) (–2)
The next three terms are 80, –160, and 320.
Holt McDougal Algebra 1

16. 9-1 Geometric Sequences
Check It Out! Example 1b
Find the next three terms in the geometric
sequence.
512, 384, 288,…
Step 1 Find the value of r by dividing each term
by the one before it.
512 384 288
The value of
r is 0.75.
Holt McDougal Algebra 1

17. 9-1 Geometric Sequences
Check It Out! Example 1b Continued
Find the next three terms in the geometric
sequence.
512, 384, 288,…
Step 2 Multiply each term by 0.75 to find the next
three terms.
288 216 162 121.5
0.75 0.75 0.75
The next three terms are 216, 162, and 121.5.
Holt McDougal Algebra 1

18. 9-1 Geometric Sequences
To find the output an of a geometric sequence
when n is a large number, you need an equation,
or function rule.
The pattern in
the table shows
that to get the
nth term,
multiply the first
term by the
common ratio
raised to the
power n – 1.
Holt McDougal Algebra 1

19. 9-1 Geometric Sequences
If the first term of a geometric sequence is a1,
the nth term is an , and the common ratio is r,
then
an = a1rn–1
nth term 1st term Common ratio
Holt McDougal Algebra 1

20. 9-1 Geometric Sequences
Example 2A: Finding the nth Term of a Geometric
Sequence
The first term of a geometric sequence is 500,
and the common ratio is 0.2. What is the 7th
term of the sequence?
an = a1rn–1 Write the formula.
a7 = 500(0.2)7–1 Substitute 500 for a1,7 for n, and
0.2 for r.
= 500(0.2)6 Simplify the exponent.
= 0.032 Use a calculator.
The 7th term of the sequence is 0.032.
Holt McDougal Algebra 1

21. 9-1 Geometric Sequences
Example 2B: Finding the nth Term of a Geometric
Sequence
For a geometric sequence, a1 = 5, and r = 2.
Find the 6th term of the sequence.
an = a1rn–1 Write the formula.
a6 = 5(2)6–1 Substitute 5 for a1,6 for n, and 2
for r.
= 5(2)5 Simplify the exponent.
= 160
The 6th term of the sequence is 160.
Holt McDougal Algebra 1

22. 9-1 Geometric Sequences
Example 2C: Finding the nth Term of a Geometric
Sequence
What is the 9th term of the geometric
sequence 2, –6, 18, –54, …?
2 –6 18 –54
The value of r is
–3.
an = a1rn–1 Write the formula.
a9 = 2(–3)9–1 Substitute 2 for a1,9 for n, and –3
for r.
= 2(–3)8 Simplify the exponent.
= 13,122 Use a calculator.
The 9th term of the sequence is 13,122.
Holt McDougal Algebra 1

23. 9-1 Geometric Sequences
Caution
When writing a function rule for a sequence with
a negative common ratio, remember to enclose r
in parentheses. –212 ≠ (–2)12
Holt McDougal Algebra 1

24. 9-1 Geometric Sequences
Check It Out! Example 2
What is the 8th term of the sequence 1000,
500, 250, 125, …?
1000 500 250 125
The value of r is .
an = a1rn–1 Write the formula.
a8 = 1000( )8–1 Substitute 1000 for a1,8 for n, and
for r.
Simplify the exponent.
= 7.8125 Use a calculator.
The 8th term of the sequence is 7.8125.
Holt McDougal Algebra 1

25. 9-1 Geometric Sequences
Example 3: Application
A ball is dropped from a
tower. The table shows Bounce Height (cm)
the heights of the balls
1 300
bounces, which form a
geometric sequence. 2 150
What is the height of the 3 75
6th bounce?
300 150 75
The value of r is
0.5.
Holt McDougal Algebra 1

26. 9-1 Geometric Sequences
Example 3 Continued
an = a1rn–1 Write the formula.
a6 = 300(0.5)6–1 Substitute 300 for a1, 6 for n, and
0.5 for r.
= 300(0.5)5 Simplify the exponent.
= 9.375 Use a calculator.
The height of the 6th bounce is 9.375 cm.
Holt McDougal Algebra 1

27. 9-1 Geometric Sequences
Check It Out! Example 3
The table shows a car’s
value for 3 years after it
Year Value ($)
is purchased. The values
form a geometric 1 10,000
sequence. How much 2 8,000
will the car be worth in
the 10th year? 3 6,400
10,000 8,000 6,400
The value of r is
0.8.
Holt McDougal Algebra 1

28. 9-1 Geometric Sequences
Check It Out! Example 3
an = a1rn–1 Write the formula.
a6 = 10,000(0.8)10–1 Substitute 10,000 for a1,10 for n,
and 0.8 for r.
= 10,000(0.8)9 Simplify the exponent.
= 1,342.18 Use a calculator.
In the 10th year, the car will be worth $1342.18.
Holt McDougal Algebra 1

29. 9-1 Geometric Sequences
Lesson Quiz: Part I
Find the next three terms in each geometric
sequence.
1. 3, 15, 75, 375,… 1875; 9375; 46,875
2.
3. The first term of a geometric sequence is 300
and the common ratio is 0.6. What is the 7th
term of the sequence? 13.9968
4. What is the 15th term of the sequence 4, –8, 16,
–32, 64…? 65,536
Holt McDougal Algebra 1

30. 9-1 Geometric Sequences
Lesson Quiz: Part II
Find the next three terms in each geometric
sequence.
5. The table shows a car’s value for three years
after it is purchased. The values form a
geometric sequence. How much will the car be
worth after 8 years?
$5570.39 Year Value ($)
1 18,000
2 15,300
3 13,005
Holt McDougal Algebra 1

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