+

Geometric Sequences & Series

EQ: How do we analyze geometric sequences & series?

M2S Unit 5a: Day 9

+Vocabulary

In a geometric sequence, the ratio of any term to the previous term is constant.

This common ratio is denoted by r.

Ex: Watch me as I work one…Tell whether the sequence

is geometric. Explain.

4,8,16,32,64,...

Geometric; r=2

+Tell whether the sequence is geometric. Explain.

2) 512, 128, 64, 8, ...

1) 1, -4, 16, -64, 256,...

; common ratio is -4yes

geometricnot

+VocabularyThe nth term of a geometric sequence with first term and common ratio r is...

1a 11 nna a r

Write a rule for the nth term of the geometric sequence.

1 14(2)nna

14) 4, 3a r

14( 3)nna13) 14, 2a r

+Write a rule and graph.

1 2(3)nna

15) One term of a geometric sequence is a 2.

The common ratio is 3. Write a rule for the

term and graph the sequence.

r

nth

Create a table of values for the sequence.Notice the points lie on an exponential curve.

n -1 0 1

-2/9 -2/3 -2na

+Write a rule and graph.

1 4(2) nna

16) One term of a geometric sequence is a 4.

The common ratio is 2. Write a rule for the

term and graph the sequence.

r

nth

Create a table of values for the sequence.Notice the points lie on an exponential curve.

n -1 0 1

1 2 4na

+Write a rule and graph.

1 3(3) nna

17) One term of a geometric sequence is a 3.

The common ratio is 3. Write a rule for the

term and graph the sequence.

r

nth

Create a table of values for the sequence.Notice the points lie on an exponential curve.

n -1 0 1

-1/3 -1 -3na

+Relationship between geometric sequences and exponential functionsThe common ratio (r) will always

represent the base (b) in an exponential function.

The first term will always be “a”

The exponent will always be “n-1”

The graph of a geometric sequence will always resemble part of an Exponential function.

+8. Pick the exponential function related to the given geometric sequence.Sequence: 4, 16, 64, 256, 1024, …

1) ( ) 2(16)na f x -=1) ( ) 4(2)nb f x -=1) ( ) 4(4)nc f x -=1) ( ) 4(2)nd f x -=

+9. Pick the exponential function related to the given geometric sequence.Sequence: 2, 6, 18, 54, …

1) ( ) 2(6)na f x -=1) ( ) 2(3)nb f x -=1) ( ) 2(2)nc f x -=1) ( ) 3(2)nd f x -=

+10. Pick the exponential function related to the given geometric sequence.Sequence: 90, 30, 10, 10/3, …

1) ( ) 90(3)na f x -=

) ( ) 90(3)nb f x =1

) ( ) 903

n

c f xæö÷ç= ÷ç ÷çè ø

11

) ( ) 903

n

d f x-æö÷ç= ÷ç ÷çè ø

+Let’s write the rule. Watch me as I work one.Write a rule for the nth term as an exponential function.

11) 972, -324, 108, -36, ... 1 927324 1927 3ar

11 11972 3nn n

na a ra

+Now you try.Give the exponential function that corresponds.

12) 6, 24, 96, 384, ...

1 624 46ar

11 16 4

nn nna a ra

13) 1, 6, 36, 216, 1296, ...

1 16 61ar

11 11 6nn nn

a a ra

+Vocabulary

An expression formed by adding the terms of a geometric sequence is called a geometric series.

The sum of the first n terms of a geometric series with common ratio r ≠ 1 is:

1 11 nn rS a r

+Find the sum of a geometric series.Watch me as I work one…

14) 7+(-21)+63+(-189)+...

Find the sum of the first 8 terms (by hand)

+Find the sum of a geometric series.Now you try.

15) 1+4+16+64+... Find the sum of the first 6 terms.

16) 1+9+81+729+... Find the sum of the first 10 terms.

+

Homework

Unit 5a Day 9 Handout