consider the following sequence: 16, 8, 4, ana… · notes 12-3 and 12-5: infinite sequences and...
TRANSCRIPT
• What kind of sequence is it?
• Find the 18th term.
• Now find the 20th, 25th, and 50th.
• So …the larger n is the more the sequence approaches what?
Consider the following sequence: 16, 8, 4, ….
Notes 12-3 and 12-5: Infinite Sequences
and Series, Summation Notation
I. Infinite Sequences and Series
A. Concept and Formula
• W=When 𝑟 < 1, as n increases, the terms of the sequence will decrease, and ultimately approach zero. Zero is the limit of the terms in this sequence.
• What will happen to the Sum of the Series?
It will reach a limit as well.
The sum, Sn, of an infinite geometric series for which 𝑟 < 1 is given by the following formula:
1
1n
aS
r
Notice that 𝑟 < 1. If 𝑟 > 1, the sum does not exist.
The series must also be geometric. Why?
Ex 1: Find the sum of the series 21 − 3 +3
7−⋯
𝑆𝑛 =𝑎11 − 𝑟
𝑆𝑛 =21
1 − (−17)
𝑟 =𝑎2
𝑎1=−3
21=−1
7
𝑆𝑛 = 18.375
Ex 2: Find the sum of the series 60 + 24 + 9.6…
𝑆𝑛 =𝑎11 − 𝑟
𝑆𝑛 =60
1 − .4
𝑟 =𝑎2𝑎1=24
60= .4
𝑆𝑛 = 100
B. Applications
Ex 1: Francisco designs a toy with a rotary flywheel that rotates at a maximum speed of 170 revolutions per minute. Suppose the flywheel is operating at its maximum speed for one minute and then the power supply to the toy is turned off. Each subsequent minute thereafter, the flywheel rotates two-fifths as many times as in the preceding minute. How many completerevolutions will the flywheel make before coming to a stop?
𝑆𝑛 =𝑎11 − 𝑟
𝑆𝑛 =170
1 −25
𝑆𝑛 = 283.3333
It makes 283 complete
revolutions before it stops.
Ex 2: A tennis ball dropped from a height of 24 feet bounces .75% of the height from which it fell on each bounce. What is the vertical distance it travels before coming to rest?
C. Writing Repeating Decimals as Fractions
• To write a repeating decimal as a fraction, start by writing it as an infinite geometric sequence.
Ex. 1: Write 0. 762 as a fraction
0. 762 =762
1000+
762
1,000,000+
762
1,000,000,000+⋯
In this series, a1=762
1000and r =
1
1000
𝑆𝑛 =𝑎11 − 𝑟=
7621000
1 −11000
=762
999=254
333
Ex 2: Write 0.123123123… as a fraction using an Infinite Geometric Series.
0. 123 =123
1000+
123
1,000,000+
123
1,000,000,000+⋯
In this series, a1=123
1000and r =
1
1000
𝑆𝑛 =𝑎11 − 𝑟=
1231000
1 −11000
=123
999
Ex 3: Show that 12.33333… = 121
3using a geometric series.
First, write the repeating part as a fraction.
0. 7 =3
10+3
100+3
1,000+⋯
In this series, a1=3
10and r =
3
10
𝑆𝑛 =𝑎11 − 𝑟=
310
1 −110
=3
9=1
3
So, 12.33333… = 121
3
II. Sigma Notation/Summation NotationIn mathematics, the uppercase Greek letter sigma is often used to indicate a sum or series. This is called
sigma notation. The variable n used with sigma notation is called the index of summation.
Summation
symbol
Upper Limit (greatest value of n)
Lower Limit (least value of n)
Expression for
the general
term
Substitute n = 1 into the equation and continue through n = 3.
(5*1 + 1) + (5*2 +1) + (5*3 + 1) =
6+11+16
33
This is read “the summation from n = 1 to 3 of 5n + 1”.
Expanded
Form
A. Writing in Expanded Form and Evaluating
Ex 1: Write the following in expanded form and evaluate:
N = 10
1st = 1
Last = 10
a1= 1 - 3 = -2
a10 = 10 - 3 = 7
Notice that since this is an arithmetic series, we could also use the formula for a finite
arithmetic series to evaluate:
25)5(5)72(2
10nS
)(2
1 nn aan
S
Expanded form:
-2 + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7
Evaluate: 25
Ex 2: Find the number of terms, the first term and the last term. Then evaluate the series:
N = 4
1st = 2
Last = 5
Expanded for:
4+9+16+25
Evaluate: 54
Note: this is NOT an
arithmetic series. You can
NOT use the formula; you
have to manually crunch out
all the values.
B. n Factorial
• As you have seen, not all sequences are arithmetic or geometric. Some important sequences are generated by products of consecutive integers. The product n(n – 2) … 3 * 2 * 1 is called n factorial and is symbolized n!.
• As a rule, 0! = 1• The table at right shows just
how quickly the numbers can grown. Copy down the first 7 rows. You will need to recognize this pattern in a subsequent example.
C. Writing a series in sigma notation
• Ex 1: 102 + 104 + 106 + 108 + 110 + 112
n = 6 terms
1st term = 1
Rule: Hmmmm. . . .
Rule = 100 + 2n
• Ex 2:Write in sigma notation: −4
1+16
2−64
6+256
24
n = 4 terms
1st term = 1
Rule: Hmmmm. . . .
The numerator has powers of 4, and the signs rotate back and forth between positive
and negative. The means the numerator should be (−1) 24𝑛
The denominator has factorials, so it should be n!
So we have
𝑛=1
4(−1)24𝑛
𝑛!
D. Applications
• Ex 1: During a nine-hole charity golf match, one player presents the following proposition: The loser of the first hole will pay $1 to charity, and each succeeding hole will be worth twice as much as the hole immediately preceding it.
• a. How much would a losing player pay on the 4th hole?
• b. How much would a player lose if he or she lost all nine holes?
• c. Represent the sum using sigma notation.
• a. Since the sequence is geometric, we can use the formula for the nth term of a geometric sequence.
an = a1rn–1
an = 1(2)4–1
an = (2)3
an = 8
The loser would have to pay $8.
• b. We can use the formula for a finite geometric series.
𝑆𝑛 =𝑎1 − 𝑎1𝑟
𝑛
1 − 𝑟
𝑆𝑛 =1−1(2)9
1−2
𝑆𝑛 =1−512
1−2
𝑆𝑛 = 511
The loser of all nine holes would have to pay $511.
• c.
•
𝑛=1
9
2𝑛−1
9 total holes
Start at the first hole
Each time is
doubled, we
start with 1$, so
20 = 1, 21 = 2,
22 = 4, etc.