Statistics and Probability - sequences-and-Series.ppsx

Series and Sequences: A Comprehensive Overview

Introduction to Sequences

Sequences are among the foundational concepts in mathematics, offering a structured way to understand patterns and ordered sets of numbers. A sequence is defined as an ordered list of numbers, each of which is called a term. These numbers follow a specific rule or pattern that determines the progression from one term to the next. The concept of sequences is not only critical in mathematics but also in various fields like computer science, physics, finance, and biology, where identifying patterns and predicting outcomes based on initial conditions are essential.

The simplest and most common examples of sequences are the arithmetic and geometric sequences. In an arithmetic sequence, each term is obtained by adding a constant difference to the preceding term. For instance, in the sequence 2, 5, 8, 11, 14..., each term is obtained by adding 3 to the previous term. Geometric sequences, on the other hand, involve a constant ratio between successive terms. For example, in the sequence 3, 9, 27, 81..., each term is obtained by multiplying the previous term by 3.

Sequences can be finite, with a specific number of terms, or infinite, continuing indefinitely. The study of sequences involves analyzing these patterns, identifying the rules that govern them, and applying these rules to solve problems or predict future terms.

Types of Sequences

Arithmetic Sequences

An arithmetic sequence is defined by a constant difference between consecutive terms, known as the common difference. If the first term of the sequence is denoted by \( a_1 \) and the common difference by \( d \), then the \( n \)-th term of the sequence can be expressed as:

\[
a_n = a_1 + (n-1) \cdot d
\]

For example, in the sequence 4, 7, 10, 13, 16..., the common difference is 3. The formula allows us to find any term in the sequence, such as the 10th term, which would be \( 4 + (10-1) \cdot 3 = 31 \).

Arithmetic sequences are widely used in various applications, from calculating the depreciation of assets over time to determining the total number of items in a series of repeated processes.

Geometric Sequences

Geometric sequences differ from arithmetic sequences in that each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. If \( a_1 \) is the first term and \( r \) is the common ratio, the \( n \)-th term of a geometric sequence is given by:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Consider the sequence 5, 10, 20, 40, 80..., where each term is multiplied by 2. The 6th term, for instance, would be \( 5 \cdot 2^{(6-1)} = 160 \).

Geometric sequences are essential in understanding exponential growth or decay, such as population growth, radioactive decay, and the compounding of interest in finance.

Fibonacci Sequence

The Fibonacci sequence is a unique sequence where each term is the sum of the two preceding ones, typically starting with 0 and 1. This sequence begins as

1. Sequences and Series

2. Sequence
•A sequence is a function whose domain is the set of positive integers or the set {1, 2, 3, …, n}
a
1
– first term, a
n
– nth term
Example 1:
Determine the first five terms of each defined sequence, and give their associated series.
(1){2 – n}
Solution: a
1 = 2 – 1 a
2 = 2 – 2 a
3 = 2 - 3
= 1 = 0 = -1
a
4 = 2 – 4 a
5 = 2 – 5
= -2 = -3
The first 5 terms are: 1, 0, -1, -2, -3
Series
-represents the sum of the terms of a sequence
Example: give the associated series of the first five terms of a sequence defined above.
S
5
= 1 + 0 + (-1) + (-2) + (-3)
= -5

3. •Example 2: {(-1)
n
}
a
1
= (-1)1 a
2
= (-1)2 a
3
= (-1)3 a
4
= (-1)4 a
5
= (-1)5
= -1 = 1 = -1 = 1 = -1
S
5 = -1 + 1 + (-1) + 1 + (-1)
= -1
•Example 2: {1 + 2n + 3n
2
}
a
1
= 1 + 2(1) + 3(1)
2
= 1 + 2 + 3(1)
= 1 + 2 + 3
= 6

S
5
= 6 + 27 + 34 + 57 + 86
= 200
a
2
= 1 + 2(2) + 3(2)
2
= 1 + 4 + 3(4)
= 1 + 4 + 12
= 17

a
3
= 1 + 2(3) + 3(3)
2
= 1 + 6 + 3(9)
= 1 + 6 + 27
= 34

a
4
= 1 + 2(4) + 3(4)
2
= 1 + 8 + 3(16)
= 1 + 8 + 48
= 57

a
4
= 1 + 2(5) + 3(5)
2
= 1 + 10 + 3(25)
= 1 + 10 + 75
= 86

4. Fibonacci sequence
The sequence {an} defined by an = an-1 + an – 2 for n where a1 = a2 = 1,
is called a Fibonacci sequence. Its terms are 1, 1, 2, 3, 5, 8, 13, …
The nth term of the Fibonacci sequence can be obtained using the Binet’s formula
The symbol means round to the nearest integer.
Example: Find f
7
using Binet’s formula.
f
7 =[
= []
= [12.98]
= 13

5. Arithmetic sequence
A sequence in which each term after the first is obtained by adding a constant (called
the common difference) to the preceding term.
If the nth term of an arithmetic sequence is a
n
and the common difference is d, then
a
n
= a
1
+ (n -1)d
The associated arithmetic series with n terms is given by

6. •How many terms are there in an arithmetic sequence with the 5 as
the first term, common difference is -3 and the last term is -76?
Solution: Given: a
1
= 5, d = -3 and a
n
= -76. Find n or the number of terms.
a
n = a
1 + (n – 1)d
-76 = 5 + (n – 1) (-3)
-76 = 5 – 3n + 3
3n = 5 + 3 + 76
3n = 84
n = 28
Therefore there are 28 terms
Example 1: Arithmetic Sequence

7. •List the first three terms of the arithmetic sequence if the 25
th
term is 35 and
the 30
th
term is 5?
Solution 1: Given: a
25 = 35, a
30 = 5
Let a
25 = a
1 and a
30 = a
6 = 5
a
6 = a
1 + (n – 1)d
5 = 35 + (6 – 1) d
5 = 35 + (5)d
5 = 35 + 5d
-5d = 35 – 5
-5d = 30
-5d/-5 = 30/-5
d = -6

Example 2: Arithmetic Sequence
Use a
25
or a
30
to get the first term
a
25
= a
1
+ (25 – 1)(-6)
35 = a
1
+ (24)(-6)
35 = a
1 + (-144)
35 + 144 = a
1
a
1 = 179
a
2
= 179 + (-6) = 173
a
3
= 173 + (-6) = 167
The first three terms are 179, 173 and 167.
You can also try using a
30 to verify these three
terms.

8. •List the first three terms of the arithmetic sequence if the 25
th
term is
35 and the 30
th
term is 5.
Solution 2: Given: a
25
= 35, a
30
= 5
a
25 = a
1 + (25 – 1)d
35 = a
1
+ 24 d
a
1
= 35 - 24d (1)
a
30 = a
1 + (30 – 1)d
5 = a
1 + 29 d
a
1
=5 - 29d (2)

Example 2: Arithmetic Sequence
Since a
1
= a
1
from equations 1 and 2
,
35 - 24d = 5 - 29d
- 24d+ 29d = 5 – 35
5d = -30
5d/5 = -30/5
d = -6
a
1 = 35 - 24d
a
1
= 35 – 24(-6)
a
1 = 35 + 144
a
1
= 179
a
2
= a
1
+ d; a
2
= 179 + (-6) ; a
2
= 173
a
3
= a
2
+ d; a
3
= 173 + (-6) ; a
3
= 167

9. •Find the sum of all positive three-digit odd integers
Solution 3: a
1
= 101; a
2
= 103, therefore d = 2; a
n
= 999
an = a1 + (n – 1) d
999 = 101 + (n-1)(2)
999 = 101 + 2n -2
999 -101 + 2 = 2n
900 = 2n
900/2 = 2n/2
n = 450
a
30
= a
1
+ (30 – 1)d
5 = a
1 + 29 d
a
1
=5 - 29d (2)

Example 3: Arithmetic Sequence
To find the sum:
Sn =
S
450 =
S
450 =
S
450
=
S
450 =

10. Geometric Sequence
•A sequence in which each term is obtained by multiplying the preceding term
by a constant (called common ratio)
•If the nth term of a geometric sequence is an and the common ratio is r, then
a
n = a
1r
n-1
• The associated geometric series with n terms is given by
S
n
= na
1
, if r = 1
1
When -1 < r< 1, the infinite geometric series has a sum given by
S =

11. •Example 1: Geometric Sequence
The seventh term of a geometric sequence is -6 and the tenth term is 162. Find the fifth term.
Solution 1: a
7
= -6; a
10
= 162; a
5
= ?
Let a
7 = a
1; a
10 = a
4
a
n = a
1r
n-1
a
4 = a
1r
4-1
162 = -6r
3

r
3
= -27
r =
r = -3
Use a
7
to get a
1
a
7
= a
1
r
7-1
-6 = a
1
(-3)7-1
-6 = a1(-3)6
-6 = a1(729)
a
1
=
a
5 = a
1r
5-1
a
5 = (-3)
5-1
a
5
= (-3)
4
a
5 = (81)
a
5
=
a
5
= -

12. •Example 1: Geometric Sequence
The seventh term of a geometric sequence is -6 and the tenth term is 162. Find the fifth
term.
Solution 2: a
7
= -6; a
10
= 162; a
5
= ?
a
7
= a
1
r
7-1
-6 = a
1r
6
a
1 =
a
10
= a
1
r
10-1
162 = a
1
r
9
a
1 =
Since a1 = a1
-6 =

= -27
= -27
r =
r = -3
a
7 = a
5r
7-5
-6 = a
5(-3)
2
-6 = a
5
(9)
a
5 =
a
5 =

13. •Example 2: Geometric Sequence
Insert three numbers (called geometric means) between 6 and 32/27, so that the five
numbers form a geometric sequence.
6, __, __, __,
a
1
a
4
Solution : If a
1
= 6 and ; a
5
=
a
5
= a
1
r
5-1
= 6r
4
x = r
4
= r
4
r
4 = ; r =
If r =
a
2
= 6 x

= 4
a
3 = 4 x
a
3
=
a
4
= x
a
4
=
6, 4, ,
If r = -
a
2
= 6 x -

= -4
a
3 = -4 x -
a
3
=
a
4
= x -
a
4
=-
6, - 4, ,

14. Harmonic Sequence
•If {a
n} is an arithmetic sequence, then the sequence with nth term b
n = is
a harmonic sequence.
•Example:
1, , , …

15. Exercises: Try to answer these numbers
(activity notebook)
1.Find the 5
th
term of the arithmetic sequence whose 3
rd
term is 35 and
whose 10
th
term is 77.
2.Suppose that the fourth term of a geometric sequence is and the sixth
term is
3.The partial sum in the arithmetic series with the first term 17 and a
common difference is 30 705. How many terms are in the series?
4. An arithmetic sequence a1, a2, ….., a100 has a sum of 15,000. Find the
first term and the common difference if the sum of the terms in the
sequence a3, a6, a9, …, a99 is 5016.
5.The sum of an infinite geometric series is 108, while the sum of the first
terms is 112. Determine the first term of this series.