Knight’s Charge
Day 6
2/2/16
Given the sequence: 24576, 12288, 6144, 3072, 1536, 768, …
1. Find the 15th term of the sequence USING A FORMULA:
2. Find the sum of the first 20 terms of the sequence USING A
FORMULA.
Homework
1. r=-2; 2, -4, 8
4. r=2.5; 125, 312.5, 781.25
9.
1 𝑛−1
10. 𝑎𝑛 = 36
;
3
13. 𝑎𝑛 = 4 −3 𝑛−1 ;
18. a) 𝑎𝑛 = 7 𝑛−1
1
𝑎1 = 36 and 𝑎𝑛 = ∙ 𝑎𝑛−1
3
𝑎1 = 4 and 𝑎𝑛 = −3 ∙ 𝑎𝑛−1
b) 16,807 people
20. 15/64
23. 5/72
28. $2516.42
32. 16, 64
33. -192, 144, -108 OR 192, 144, 108
40. 1092
41. 85.5
“Live in the present
to learn of the past
and be a part of the future.”
-Jaleel Mott
SEQUENCE:
Infinite Geometric Series

An infinite geometric sequence is one whose terms go on
forever.
r =2.... The terms in the sequence are getting



Example: 1, 2, 4, 8, 16, 32, …
Example: 32, 16, 8, 4, 2, 1, …
larger and larger.
r =12 .... The terms in the sequence are getting
closer and closer to 0.
An infinite geometric series is the sum of the terms of an
infinite geometric sequence. Since the terms of the related sequence are

Example: 1 + 2 + 4 + 8 + … getting larger and larger, the sum of the terms has
no specific sum (we say the series DIVERGES).
 Example: 32 + 16 + 8 + 4 + ⋯ Since the terms of the related sequence are
getting closer to 0, the sum of the terms is
approaching a specific number (we say the series
CONVERGES). What does the sum converge
to?

The sum of a finite geometric sequence with common ratio r
𝑎1
is S =
where 𝑟 < 1.
1−𝑟
Infinite Geometric Series
Converge or Diverge?

An infinite geometric series CONVERGES if 𝑟 < 1.
𝑎
We use the formula 𝑆𝑛 = 1 to find the value it
1−𝑟
converges to.

An infinite geometric series DIVERGES if… 𝑟 ≥ 1.
Infinite Geometric Series
Example: Calculate the sum of the sequence:
{2, 4, 8, 16,…}

This is a geometric series with r=2. Since r>1, the series
DIVERGES and therefore there is no sum.
Infinite Geometric Series
Example: Calculate the sum of the sequence:
{16, 8, 4, 2, …}


This is a geometric series with r=0.5. Since 𝑟 < 1, the
series CONVERGES and therefore we can find its sum.
We’ll use the formula 𝑆 =
𝑎1
.
1−𝑟
16
𝑆=
1 − 0.5
𝑆 = 32
Infinite Geometric Series
Example: Calculate the sum of the sequence:
{16, -8, 4, -2, …}


This is a geometric series with r=-0.5. Since 𝑟 < 1, the
series CONVERGES and therefore we can find its sum.
We’ll use the formula 𝑆 =
𝑎1
.
1−𝑟
16
𝑆=
1 − (−0.5)
𝑆 = 10. 6
Infinite Geometric Series
Example: Find the common ratio r given 𝑎1 =
1 and 𝑆 = 1.25.

We’ll use the formula 𝑆 =
𝑎1
.
1−𝑟
1
1.25 =
1−𝑟
1.25 1 − 𝑟 = 1
* Distribute 1.25
1.25 − 1.25𝑟 = 1
* Solve for r.
𝑟 = 0.2
Infinite Geometric Series
* Multiply by 1-r
Example:

A tennis ball dropped from a height of 30 feet bounces
40% of the height from which it fell on each bounce.
What is the vertical distance it travels before coming to
rest?
Infinite Geometric Series
Example:

Write 0. 579 as a fraction.
Infinite Geometric Series
Example:

Write 0. 83 as a fraction.
Infinite Geometric Series
BINGO

Fold a sheet of paper in half FOUR times.
(Think of this as a geometric sequence where 𝑎1 = 1 and r=2. How many
“squares” should you have after folding a piece of paper 4 times? )

Place one of the following in each of the squares.
Converges
8
45
64
𝟏
-1
405
-0.25
2
𝟐
𝟐𝟓
-268.8
𝟗
27
12
𝟐
𝟓
𝟑
𝟏𝟏
Diverges
𝟏
𝟐
5000
156.25
No Sum
VICTORY LAP
Determine whether or not the following series
CONVERGES or DIVERGES. If it converges, find the sum.
1.
2 − 4 + 8 − 16 + ⋯
2.
16 + 8 + 4 + 2 + ⋯
Find the sum of the infinite
geometric series described.
Find the sum of the infinite
geometric series.
Find the sum of the infinite
geometric series.
Determine the common ratio of
the infinite geometric series.
Determine the common ratio of
the infinite geometric series.
Does the series converge or diverge?
Does the series converge or diverge?
Determine the common ratio of
the infinite geometric series.
Find the sum of the series:
20
15 + 10 +
+⋯
3
Find the sum of the series:
1000 + 800 + 640 + ⋯
Find the sum of the series:
1
10
+
1
20
+
1
+…
40
Find the sum of the series:
540 − 180 + 60 − 20 + ⋯
Homework

“11-3 Practice” worksheet
#1-21 ODD

Unit 1 Test on FRIDAY!