Standard Accessed: Students will analyze sequences, find sums of series, and use recursive rules.

Warm-up Check

Find each sum.

1. 100 + 50 + 25 175

2. 100 + 50 + 25 +

3. 100 + 50 + 25 + +

4. 100 + 50 + 25 + + +

187.5

193.75

196.875

Big Idea• When does an infinite

geometric series have a sum, and when does it not have a sum?

If for an infinite geometric series, the series has the sum If for an infinite geometric series, the series has no sum.

Vocabulary

The sum of the first n terms of an infinite series. The partial sums of an infinite geometric series may approach a limiting value.

Find Partial SumsEXAMPLE 1

SOLUTION

Consider the infinite geometric series 2 + . Find and graph the partial sums for n = 1, 2, 3, 4, and 5. Then describe what happens to as n increases.

𝑺𝟏=𝟐 ,𝑺𝟐≈𝟐 .𝟔𝟕 ,𝑺𝟑≈𝟐 .𝟖𝟗 ,𝑺𝟒≈𝟐 .𝟗𝟔 ,𝑺𝟓≈𝟐.𝟗𝟗

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

1.5

2

2.5

3

3.5

2

2.672.89 2.96 2.99

Geometric Infinite Series

N - Term

𝑎_n Value

*Note: As n increases, appears to approach 3.

Find the sum of the infinite geometric series.EXAMPLE 2

SOLUTION

(𝑎 )∑𝑖=1

∞

3( 14 )

𝑖−1

;

(𝑏)− 12+

14−

18+

116

+…

;

EXAMPLE 3

SOLUTION

D) Does not Exist, r

*Check by graph

What is the sum of the infinite series.

a) -2

b) -

c) 2

d) Does not exist

∑𝑖=1

∞

1 ( 32 )

𝑖− 1

;𝑺=𝒂𝟏

𝟏−𝒓= 1

1−32

=−2

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

1

2

3

4

5

6

11.5

2.25

3.375

5.0625

Geometric Infinite Series

N - Term

𝑎_n Value

Clifford plays ball:

EXAMPLE 4 Use an Infinite Geometric series in real life

Clifford drops his rubber ball from the roof of KIS, about 60 feet. Each bounce takes it to 2/3 of its previous height. What is the total vertical distance the ball travels?.

; 𝑺=

60

𝟏−( 23 )

=𝟏𝟖𝟎−𝟔𝟎=𝟏𝟐𝟎

(𝟏𝟐𝟎∗𝟐 )+𝟔𝟎=𝟑𝟎𝟎 𝒇𝒕

EXAMPLE 5

SOLUTION

Write 5.146146146… as a fraction in lowest terms.

Write a repeating decimal as a fraction.

5+∑𝑖=1

∞

146 (0.001 ) (0.001 )𝑖− 1

5.146146…

¿5+( 0.1461−0.001 )=𝟓 𝟏𝟒𝟔

𝟗𝟗𝟗𝒐𝒓 𝟓𝟏𝟒𝟏

𝟗𝟗𝟗

Lesson 7.4Analyze Geometric Sequences & Series

Homework:Honors – HomeworkAcademic – Homework