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
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