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NUMERICAL SEQUENCES v.07
Many times we are concerned about listing of numbers. The listing can be finite or
infinite. The listing 81, 2, 3, 4, ....< is infinite, while 82, 4, 6< is
finite. A listing of numbers where the order matter is called a sequence (order matters:
first, second, third, etc).
Since order matters, 81, 2, 3, 4, ....< ¹ 82, 1, 3, 4, ...<, and
81, 2, 4< ¹ 82, 1, 4< as sequences. In this introduction you will learn about
multiplicative and additive sequences.
EXAMPLE 1 (Multiplicative sequence)
Consider the case of the "crazy ball". This ball gains height at each bounce (wow!).
The ball is dropped from a height of 3 meters and at each bounce its height doubles.
The first five heights of the "crazy ball" are 83, 6, 12, 24, 48, ...<
The list can be represented on the number line.
3
6
24
12
The values of this list grow without a boundary. What does it mean?
Notice that each term in the sequence is obtained by multiplying the previous term by
2. For this reason this is called a multiplicative sequence or geometric sequence with 2
as the multiplicative factor.
A listing can be interpreted as a function, where the inputs indicate the position of the
number in the list, and the outputs are the values in the list. Normally, the position zero
is assigned to the first term. With this convention, the first five terms of the listing
above are given in the table below as a function.
n
0 1 2 3 4
aHnL 3 6 12 24 48
NOTATION
The term in the sequence in the nth position is denoted as aHnL or simply an for
short.
So far there are two ways to represent a sequence graphically . Using a number line and
as function. Both representations are shown
3
6
12
24
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Sequences Introduction v07.nb
3
6
24
12
aHnL
40
30
a4
20
ƒƒ aaaƒƒ
a3
10
a2
a
a11
a0
1
2
3
4
n
Now, let's find algebraic representations for the sequence.
The sequence was described as starting at 3 (initial condition), and then each term after
the first is obtained by multiplying the previous term by 2 (recursive relation). These
two conditions are summarized below. It is called the recursive definition of the
sequence
∂
a0 = 3
an+1 = 2 * an n ³ 1
Using the recursive definition, the terms in the sequence can be obtained one after the
other. If you now a0 that value is used to produced a1 , if you know a1 that value is
used to produced a2 , and so forth.
Sometimes the term of the sequence can be determined by knowing his position in the
sequence (the value of n). To be able to do it there is the need to find an explicit formula to represent this sequence. Below, it is illustrated how to find an explicit formula
for a multiplicative sequence.
From writing the first few terms using the recursive definition, a pattern of formation
is observed. Notice that after the value of n (subindex is known), the term can be calculated.
Sometimes the term of the sequence can be determined by knowing his position in the
sequence
(the value
Sequences
Introduction
v07.nbof n). To be able to do it there is the need to find an explicit for- 3
mula to represent this sequence. Below, it is illustrated how to find an explicit formula
for a multiplicative sequence.
From writing the first few terms using the recursive definition, a pattern of formation
is observed. Notice that after the value of n (subindex is known), the term can be calculated.
a0 = 3 * 20 , a1 = 3 * 21 , a2 = 3 * 22 , ..., a10 = 3 * 210 , ...
The pattern is an = 3 * 2n , for n = 0, 1, 2, 3, ... .
n ¥
The sequence is also written as 8an <¥
n=0 = 83 * 2 <n=0 . This expression corresponds to
exponential function but with domain non-negative integers.
EXERCISE 1
1. Find the recursive definition and the explicit formula for each of the multiplicative
sequences below:
a. 92,
2
2
, 29 , 27
,
3
....=
b. 8-1, 2, -4, 8, -16<
c. 80.1, 0.01, 0.001, ....<
2. Discuss whether or not the following comment is true or false. Zero could be a term
of a multiplicative sequence that has at least one term that is not zero.
EXAMPLE 2 (Additive sequence)
Another sequence that will appear often in our discussion is the arithmetic sequence.
The main feature of this sequence is that each term is obtained from the previous one
by adding a constant.
The sequence 8-2, 4, 10, 16, ...<is an arithmetic or additive sequence that
starts at -2 and each term, after the first one, is obtained by adding 6 to the previous
term. This is the recursive definition for the sequence:
b0 = -2
∂
bn+1 = bn + 6, n ³ 0
An explicit formula can also be obtained by looking at few terms obtained by the recursive definition. Let's look at the first ten terms for a pattern of formation.
starts at -2 and each term, after the first one, is obtained by adding 6 to the previous
term. This is the recursive definition for the sequence:
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Sequences Introduction v07.nb
b0 = -2
∂
bn+1 = bn + 6, n ³ 0
An explicit formula can also be obtained by looking at few terms obtained by the recursive definition. Let's look at the first ten terms for a pattern of formation.
b0 = -2, b1 = -2 + 6 * 1, b2 = -2 + 6 * 2,
b3 = -2 + 6 * 3, ... b10 = -2 + 6 * 10 ...
The sequence is described by the formula bn = -2 + 6 n, n ³ 0
Notice that this formula is the same as the one for linear functions, but the domain are
just positive integers.
In examples 1 and 2 both sequences were infinite, since they have an infinite number of
terms. However, the sequences can also be finite by restricting the domain.
For instance the arithmetic sequence 8-2, 4, 10, 16< is defined by the formula
cn = -2 + 6 n, 0 £ n £ 3
The multiplicative sequence 82, 6, 18, 54, 162, 486< has an explicit formula
bn = 2 * 3 n , 0 £ n £ 5
Of course, there are sequences which are neither multiplicative nor additive. Some
times they can be defined either recursively or explicitly, but not always.
For instance, the sequence 9 12 ,
2
, 34 ,
3
...= is neither multiplicative nor additive.
Why? However, it can be described by the formula rn =
mula for this sequence is rn =
n+1
,
n+2
n
,
n+1
n ³ 1. Another for-
n ³ 0. As we will discuss later, the can start at
any value.
EXERCISE 2
For each of the sequences (a)-(f) :
I. Identify which ones are arithmetic or geometric sequences and justify your answer.
Find their recursive definition, and explicit formulas for the ones that are arithmetic or
geometric.
II. Find either the recursive definition or the explicit formula to describe each of the
sequences which are neither arithmetic nor geometric.
III. From the list of values of the sequence indicate whether the values of the sequence
are "approaching" to a number.
a. 91, 12 , 13 , 14 , ...=
b. 8-1, 0, 1, 2, 3, ...<
geometric.
II. Find either the recursive definition or the explicit formula to describe each of the
Sequences Introduction v07.nb
sequences which are neither arithmetic nor geometric.
III. From the list of values of the sequence indicate whether the values of the sequence
are "approaching" to a number.
a. 91, 12 , 13 , 14 , ...=
b. 8-1, 0, 1, 2, 3, ...<
c. 8-2, 1, 4, 7, ....<
d. {-2, 1, 4, 7}
e. 9- 32 , - 43 , - 54 , - 65 , ...=
f. 81, 1, 1, ....<
g. 91, - 12 , 13 , - 14 , 15 , ...=
f. 80.9, 0.99, 0.999, 0.9999, ...<
EXERCISE 3
Graph each of the sequences in exercise 2 as functions (make sure you know how to
use technology to do it),
a. What is the domain of each function?
b. What is the range of each function?
c. Are the values in the range approaching to any particular number? Compare your
answer to exercise 2 (III).
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