# Alg 1B 8-6 Geometric Sequences by hedongchenchen

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• pg 1
Lesson 8-6
Geometric Sequences,
page 424

Objectives: To form geometric
sequences and to use formulas when
describing geometric sequences.
Why should we learn this?

One    real-world connection is
to find the height of a ball
after a number of bounces.
(example 5)
REVIEW from Chapter 5

 Sequence – a number pattern
 Term – each number in a sequence
 Arithmetic sequence – add a fixed
number to each term to find next term
 Common difference – fixed number
Definitions

   Geometric sequence – multiply a term
by a fixed number to find next term

   Common ratio – fixed number being
multiplied in a geometric sequence
Find the common ratio of each sequence.

a. 3, –15, 75, –375, . . .
3                –15              75           –375

(–5)           (–5)            (–5)
The common ratio is –5.
3       3   3
b. 3, 2 , 4 , 8 , ...
3            3                3
3                   2            4                8

1                1                1
   2
   2
   2
1
The common ratio is 2 .

8-6
Example 1, page 424

Find the common ratio.    On your own…
a) 750, 150, 30, 6, …    C) 4, 6, 9, 13.5, …

   THINK, PAIR,
SHARE…
b) -3, -6, -12, -24, …
HINT

   When trying to find the common ratio,
divide a term by the previous term.
Repeat to find the common ratio.

   Example: -2, 8, -32, 128, …

8 / -2 = -4           -32 / 8 = -4
64 / -32 = -4         Common ratio = -4
Find the next three terms of the sequence 5, –10, 20, –40, . . .

5       –10           20      –40

(–2)     (–2)        (–2)

The common ratio is –2.

The next three terms are –40(–2) = 80, 80(–2) = –160, and –160(–2) = 320.

8-6
Example 2, page 425
 Find the next two         C) 1.1, 2.2, 4.4, 8.8
terms in the
sequence.
 A) 1, 3, 9, 27,…

   B) 120, -60, 30, -
15,…
Geometric or Arithmetic?

   Is each term being multiplied by fixed number?
If so, what kind of sequence is it?

   Does each term have a fixed number being
added? If so, what kind of sequence is it?

   If not, it’s neither geometric or arithmetic.
Determine whether each sequence is arithmetic
or geometric.

a. 162, 54, 18, 6, . . .

62        54           18         6

1            1          1
3            3          3

The sequence has a common ratio.

The sequence is geometric.

8-6
(continued)

b. 98, 101, 104, 107, . . .

98        101        104         107

+3         +3         +3

The sequence has a common difference.

The sequence is arithmetic.

8-6
Example 3, page 425
Arithmetic, Geometric or Neither?

   a) 2, 4, 6, 8,…        c) 1, 3, 5, 7,…

   b) 2, 4, 8, 16, …
USING A FORMULA

A(n) = a • r n – 1

 A(n) = nth term
 a = first term
 r = common ratio
 n = term number
Find the first, fifth, and tenth terms of the sequence that has the rule
A(n) = –3(2)n – 1.

first term: A(1) = –3(2)1 – 1 = –3(2)0 = –3(1) = –3

fifth term: A(5) = –3(2)5 – 1 = –3(2)4 = –3(16) = –48

tenth term: A(10) = –3(2)10 – 1 = –3(2)9 = –3(512) = –1536

8-6
Example 4, page 426

Find the first, sixth, and twelfth terms of each sequence.
   A) A(n) = 4 • 3n-1            B) A(n) = -2 • 5n-1
Suppose you drop a tennis ball from a height of 2 meters.
On each bounce, the ball reaches a height that is 75% of its previous
height. Write a rule for the height the ball reaches on each bounce. In
centimeters, what height will the ball reach on its third bounce?

The first term is 2 meters, which is 200 cm.

Draw a diagram to help understand the problem.

8-6
(continued)
The ball drops from an initial height, for which there is no bounce. The
initial height is 200 cm, when n = 1. The third bounce is n = 4. The
common ratio is 75%, or 0.75.
A rule for the sequence is A(n) = 200 • 0.75n – 1.
Use the sequence to find the height of
A(n) = 200 • 0.75n – 1
the third bounce.
Substitute 4 for n to find the height of
A(4) = 200 • 0.754 – 1
the third bounce.
= 200 • 0.753          Simplify exponents.
= 200 • 0.421875       Evaluate powers.
= 84.375               Simplify.

The height of the third bounce is 84.375 cm.

8-6
REAL-WORLD CONNECTION
Example 5, page 426
   You drop a basketball from a height of 2 meters. Each curved
path has 56% of the height of the previous path. Using the height
in centimeters, write a rule for the sequence. What height will the
basketball reach at the top of the fourth path (where n = 4)?
Round to the nearest tenth of a centimeter.
Summary

   What did you learn today?
SUMMARY

 Look for a pattern.
 Find the common difference (arithmetic)
or ratio (geometric).
 Substitute for n to find a specific term
when given a formula for a sequence.
ASSIGNMENT

#8-6, page 427

1-43 odd
51-71 odd

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