13
Sequences Chapter and Series
13.1 Sequences
Key Points
Definition of a sequence
Arithmetic and geometric sequences
Sequences
Examples
Finite infinite
Notation
Notation for Sequences and Examples
We denote the terms of a sequence by
a1, a2, a3, . . . , an, . . .
so that a1 is the first term, a2 is the second term, and so on. We
use an to denote the nth or general term of the sequence. If there
is a pattern in the sequence, we may be able to find a formula for
an.
Example 1
(a) 0, 1, 4, 9, 16, 25, . . . is the sequence of squares of integers.
(b) 2, 4, 8, 16, 32, . . . is the sequence of positive integer powers of 2.
(c) 3, 1, 4, 1, 5, 9, . . . is the sequence of digits in the decimal expansion
of .
(d) 3.9, 5.3, 7.2, 9.6, 12.9, 17.1, 23.1, 38.6, 50.2 is the sequence of U.S.
population figures, in millions,for the census reports (1790 -1880).
(e) 3.5, 4.2, 5.1, 5.9, 6.7, 8.1, 9.4, 10.6, 10.1, 7.1, 3.8, 2.1, 1.4, 1.1 is the
sequence of pager subscribers in Japan, in millions, from the years
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1989 -2002.
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4th
Notation for Sequences and
Examples
Like Example 3
List the first 5 terms of the sequence an = f(n),
where f(x) = 400 20x.
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Arithmetic Sequences
Example 4 Which of the following sequences are
arithmetic?
(a) 9, 5, 1,3,7
(b) 3, 6, 12, 24, 48
(c) 2, 2 + p, 2 + 2p, 2 + 3p (d) 10, 5, 0, 5, 10
Solution
(a) Each term is obtained from the previous term by
subtracting 4. This sequence is arithmetic.
(b) This sequence is not arithmetic: each terms is twice the
previous term. The differences are 3, 6, 12, 24.
(c) This sequence is arithmetic: p is added to each term to
obtain the next term.
(d) This is not arithmetic. The difference between the
second and first terms is 5, but the difference between the
fifth and fourth terms is 5.
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Find the 8th term and the nth term of
the arithmetic sequences.
Arithmetic Sequences
For n 1, the nth term of an arithmetic
sequence is
an = a1 + (n 1)d,
where a1 is the first term, and d is the
difference between consecutive terms.
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Geometric Sequences
Example 6
Which of the following sequences are geometric?
(a) 5, 25, 125, 625, . . .
(c) 12, 6, 4, 3, . . .
(b) 8, 4,2, 1,1/2 , . . .
Solution
(a) This sequence is geometric. Each term is 5 times the
previous term. Note that the ratio of any term to its
predecessor is 5.
(b) This sequence is geometric. The ratio of any term to the
previous term is 1/2 .
(c) This sequence is not geometric. The ratios of
successive terms are not constant: a2/a1 = 6/12 = 1/2, but
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a3/a2 = 4/6 = 2/3.
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Find the 8th term and the nth term of
the geometric sequences.
Geometric Sequences
For n 1, the nth term of a geometric
sequence is
an = a1rn1,
where a1 is the first term, and r is the
ratio of consecutive terms.
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28,32
Chapter 13
Sequences and Series
13.2 Defining Functions Using
Sums: Arithmetic Series
Key Points
Arithmetic series as models of growth
Summation notation
Consider a theater with 40 rows of
seats. Suppose that the first row
contains 18 seats and the number of
seats increases by two per row. What is
the total number of seats?
Series and Partial Sums
The sum of the terms of a sequence is called a
series. We write Sn for the sum of the first n terms
of the sequence, called the nth partial sum. We see
that Sn is a function of n, the number of terms in
the partial sum. For the sequence {a1, a2, a3, ,
an, }
,
S1 = a1
S2 = a1 + a2
S3 = a1 + a2 + a3
Sn = a1 + a2 + a3+ + aFunctions Modeling
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General Formula for the
Sum of Any Arithmetic Series
S n = a1 + a2 + a3 + + an = (a1 + an ) + (a2 + an 1 ) + (a3 + an 2 ) +
(
12 n ) pairs
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Summation Notation
The symbol is used to indicate addition. This
symbol, pronounced sigma, is the Greek capital
letter for S, which stands for sum. Using this
notation, we write
n
ai to stand for the sum a1 + a2 + + an
i =1
The tells us we are adding some numbers. The
ai tells us that the numbers we are adding are
called a1, a2, and so on. The sum begins with a1
and ends with an because the subscript i (at the
bottom of the sign) begins at 1 and ends at n (at
the top of the sign).
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Using the Sigma Notation
6,10, 14, 20, 24, 28, 34
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Chapter 13
Sequences and Series
13.3 Finite Geometric Series
Key Points
Geometric series as models of growth
Bank Balance
A person deposits $2000 every year in an IRA that pays 6%
interest per year, compounded annually. After the first deposit (but
before any interest has been earned), the balance in the account
in dollars is B1 = 2000. After 1 year has passed, the first deposit
has earned interest, so the balance becomes 2000(1.06)
dollars. Then the second deposit is made and the balance
becomes
B2 = 2nd deposit + 1st deposit with interest
= 2000 + 2000(1.06) dollars.
After 2 years have passed, the third deposit is made, and the
balance is
B3 = 3rd dep+2nd dep with 1 yr interest +1st dep with 2 yrs interest
= 2000 + 2000(1.06)+ 2000(1.06)2.
Let Bn be the balance in dollars after n deposits. Then we see that
Bn = 2000 + 2000(1.06)+ 2000(1.06)2 + + 2000(1.06)n1.
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Geometric Series
A finite geometric series is a sum of the
form
Sn = a + ar + ar2 + + arn1
or using sigma notation with first term a and
common ratio r:
n
S n = ar i 1.
i =1
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The Sum of a Geometric Series
The
The sum of a finite geometric series of n terms is given by
a (1 r n )
2
n 1
S n = a + ar + ar + + ar =
for r 1.
(1 r )
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The Sum of a Geometric Series
Like Example 2
Use the formula for the sum of a geometric series to solve
Example 1 with the bank deposits (find B7 and B30).
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12,14,6, 4,18,2
Chapter 13
Sequences and Series
13.4 Infinite Geometric Series
Key Points
Infinite geometric series
Long-term Drug Level in the Body
In the drug dosage example, what happens to the drug level in the
patients body? To find out, we calculate the drug level after
10, 15, 20, and 25 injections.
Using the formula based on 20mg of a drug, with a metabolic halflife of one day, injected each day into a person, the amount of the
drug present after n days was found to be
Qn = 20 (1 ( )n) / (1 ).
After the 10th, 15th, 20th, and 25th injection, the drug level in the
patients body is given by
Q10 = 20 (1 ( )10) / (1 ) = 39.960938 mg,
Q15 = 20 (1 ( )15) / (1 ) = 39.998779 mg,
Q20 = 20 (1 ( )20) / (1 ) = 39.999962 mg,
Q25 = 20 (1 ( )25) / (1 ) = 39.999999 mg.
The drug level appears to approach 40 mg
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The Sum of an Infinite Geometric Series
Consider the geometric series Sn = a + ar + ar2 + + arn1. In
general, if |r| < 1, then rn 0 as n , so
a (1 r n )
a (1 0)
a
Sn =
=
as n .
1 r
1 r
1 r
Thus, if |r| < 1, the partial sums Sn approach a finite value, S, as n
. In this case, we say that the series converges to S.
For |r| < 1, the sum of the infinite geometric series is given by
For
a
2
n 1
i
S = a + ar + ar + + ar + = ar =
.
1 r
i =0
For |r| 1, the value of Sn does not approach a fixed number as n
Functions
, so we say that the infinite series does not converge. Modeling
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Infinite Geometric Series
8,10,14,16,22