SEQUENCES AND SERIES
Sequences and Series
Pattern
and Sequences
Summation Notation
Arithmetic Sequences
Geometric Sequences
Pattern and Sequences
Learning Objectives:
 Define
sequence, finite and infinite
sequence;
 List the next few terms given several
consecutive terms of a sequence; and
 Solve problems involving sequences.
A sequence is a function whose domain is
the set of positive integers. It also means an
ordered list of numbers. Ex. 1, 3, 5, 7, 9, …
A sequence is infinite if its domain is the set
of positive integers without a last term,
1, 2, 3, 4, 5, … . The three dots show that the
sequence goes on and on indefinitely.
A sequence is finite if its domain is the set of
positive integers 1, 2, 3, 4, 5, … , 𝑛 which has a last
term, n.
Each number in a sequence is called a term.
Ex.: 5, 15, 25, 35, 45.
The notation 𝑎1 , 𝑎2 , 𝑎3 , …, 𝑎𝑛 is used to denote the different
terms in a sequence.
The expression 𝑎𝑛 is referred to as the general or nth term
of the sequence.
Example:
In the sequence 1, 3, 6, 10, 15, … we can denote the terms as
follows:
𝑎1 = 1, 𝑎2 = 3, 𝑎3 = 6, 𝑎4 = 10, 𝑎5 = 15
Example 1: Write the five terms of a sequence described by the
general term 𝑎𝑛 =3n+2
𝑎𝑛 = 3n+2
𝑎1 = 3(1)+2= 5
𝑎2 = 3(2)+2= 8
𝑎3 = 3(3)+2= 11
𝑎4 = 3(4)+2= 14
𝑎5 = 3(5)+2= 17
Therefore, the first five terms are 5, 8, 11, 14, and 17
Example 2:
Write the first five terms of 𝑎𝑛 = 2 3𝑛−1 .
𝑎𝑛 = 2 3𝑛−1 .
𝑎1 = 2 31−1 = 2 30 = 2
𝑎2 = 2 32−1 = 2 31 = 6
𝑎3 = 2 33−1 = 2 32 = 18
𝑎4 = 2 34−1 = 2 33 = 54
𝑎5 = 2 35−1 = 2 34 = 162
∴ the first five terms are 2, 5, 18, 54, and 162.
Summation Notation
- a simple method for indicating the sum of a
finite (ending) number of terms in a sequence. This
involves the greek letter sigma, 𝚺. The lower number is
the lower limit of the index (the term where the
summation starts), and the upper number is the upper
limit of the summation (the term where the summation
ends). Consider
7
(2𝑘 + 3)
𝑘=2
7
𝑘=2(2𝑘
+ 3)= 2 2 + 3 + 2 3 + 3 + 2 4 + 3 + 2 5 + 3
+ 2 6 +3 + 2 7 +3
= 7 + 9 + 11 + 13 + 15 + 17
=72
Example 1: Write out the terms of the following sums; then
compute the sum.
5
3𝑖
𝑖=0
1.)
5
𝑖=0 3𝑖
= 3(0) +3(1) +3(2) +3(3) + 3(4) + 3(5) = 45
6
2𝑛
2.)
6
𝑛
𝑛=0 2
𝑛=0
1
2
= 20 + 2 + 2 + 23 +24 + 25 + 26 = 127
5
3.)
3𝑚 + 2
𝑚+1
𝑚=1
3𝑚+2
5
8
11
14
17
5
=
+
+
+
+
𝑚=1 𝑚+1
2
3
4
5
6
150+160+165+168+170
=
60
813
= 60
271
= 20
Example 2: Use sigma notation to express each series.
1. 8 + 11 + 14 + 17 + 20
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
= 8 + (n - 1) 3
= 3n + 5
Since there are five terms, the given series can be written as
5
5
𝑎𝑛 =
𝑛=1
(3𝑛 + 5)
𝑛=1
2. )
2
3
3
2
9
4
-1+ - +
27
8
81
16
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
2
3 𝑛−1
= (- )
3
2
Since there are six terms in the given series, the sum can be
written as
6
6
𝑎𝑛 =
𝑛=1
𝑛=1
2 −3
3 2
𝑛−1
Arithmetic Sequences
Learning Objectives:
1.
2.
3.
Illustrate an arithmetic sequence
Determine the nth term of
arithmetic sequence
Determine the arithmetic means
an
Arithmetic Sequence
An arithmetic sequence or arithmetic progression is a
sequence in which each term is obtained from the preceding
term by adding a common difference. Its general term is
described by
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
The number d is called the common difference.
Example 1: Find the common difference in each of the
following arithmetic sequences. Then express each sequence
in the form 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 and find the 20th term of the
sequence.
1.) 1, 5, 9, 13, 17, …
2.)
5 3
1 1 3
- , - ,− , , ,…
8 8
8 8 8
1.) 1, 5, 9, 13, 17, …
Since d= 4,
𝑎20 = 1 + 20 − 1 4
= 1+(19) 4
= 77
Therefore, the 20th term of the sequence is 77.
5
8
3
8
1 1 3
8 8 8
1
4
2.) - , - , − , , , …
Since d =
5
8
𝑎20 = − + 20 − 1
Therefore,
20th
1
4
=−
5
8
1
4
+ (19) = −
term of the sequence is
5
8
33
.
8
+
19
4
=
−5+38
8
=
33
8
3.) Find the 12th term in the sequence 11x-5, 14x-2, 17x+1, …
Given:
𝑎𝑛 = 12; 𝑎1 = 11𝑥 − 5; n=12; d = 3x + 3
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎12 = 11𝑥 − 5 + 12 − 1 (3𝑥 + 3)
𝑎12 =11x-5+ (11)(3x+3)
𝑎12 = 11x-5 + (33x+33)
𝑎12 = 11x-5+33x+33
𝑎12 = 44x+28
Thus, the 12th term is 𝑎12 = 44x +28
Arithmetic Series
- the sum of the terms in an arithmetic
sequence given by:
𝒏
𝒔𝒏 =
𝒂 𝟏 + 𝒂𝒏
𝟐
Where:
𝑠𝑛 is the sum of the terms;
𝑎1 is the first term;
n is the number of terms.
Example 1:
Find the sum of the first 10 terms in the sequence 9,16, 23, 30, …
Find 𝑎𝑛 first.
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
𝑎10 = 9+(10 - 1)7
𝑎10 = 9+(9)7
𝑎10 = 9+63
𝑎10 = 72
Find the sum using the defined formula:
𝒏
𝒔𝒏 =
𝒂 + 𝒂𝒏
𝟐 𝟏
10
𝑠10 =
9 + 72
2
𝑠10 = 5 81
𝑠10 = 405
Example 2:
The seat plan of a concert venue is arranged in a sequence. The first row
has 5 seats. The seats for the succeeding rows are 3 more than the
previous. If there are 42 rows, what is the seating capacity of the concert
venue?
Solution:
Given: 𝑎1 = 5; n = 42; d = 3
Find 𝑎𝑛 :
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
𝑎42 = 5 + 42 − 1 3
𝑎42 = 5 + 41 3
𝑎42 = 5 + 123
𝑎42 = 128
Then, find the seating capacity.
𝒏
𝒂𝟏 + 𝒂𝒏
𝟐
42
𝑠42 =
5 + 128
2
𝑠42 = 21 133
𝑠42 = 2,793
𝒔𝒏 =
Thus, the seating capacity of the concert venue is 2,793.
Arithmetic Mean
- also known as average, is a number calculated by adding two
numbers a and b and dividing by the number of terms in the set denoted by
𝑎+𝑏
2
Example 1:
Insert an arithmetic mean between 15 and 27.
solution:
15+27
2
= 21
Example 2:
Insert four arithmetic means between -6 and 14
Solution: solve for d
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
14 = -6 + (6-1)d
14 + 6 = 5d
20 = 5d
d=4
Adding the common difference, 4, with first term, -6, is the first arithmetic
mean.
Thus, the arithmetic means between -6 and 14 are -2, 2, 6, 10.
Geometric Sequences and Series
Objectives:
1.) Determine the nth term of a geometric sequence
2.) Determine the geometric mean
3.) Solve problems involving geometric sequence
4.) Differentiate a geometric sequence from an
arithmetic sequence
Geometric Sequence
- A geometric sequence or progression is a sequence in which
each term is obtained from the preceding term by multiplying a common
ratio.
- The general formula for finding the nth term of a geometric
sequence is 𝑎𝑛 = 𝑎1 𝑟 𝑛−1
Example 1:
Find the eighth term of the geometric sequence of numbers 5, 10, 20, …
Solution:
Given: 𝑎𝑛 = 𝑎8 ; 𝑎1 = 5; n = 8
5 ⦁ r = 10
r=
10
5
r=2
Use the formula
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
𝑎𝑛 = 5(2)8−1
𝑎𝑛 = 5(2)7
𝑎𝑛 = 5 128 = 640
Thus, the eight term is 640
Example 2:
7
7
Given the sequence of numbers -21, -7, - , - , find the seventh
3 6
term.
Solution:
Given: 𝑎𝑛 = 𝑎7 ; 𝑎1 = -21; n= 7; r =
7 1
=
21 3
Use the formula:
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
1 7−1
𝑎7 = −21( )
3
6
1
𝑎7 = −21
3
1
𝑎7 = −21
729
𝑎7 = −
21
729
=−
7
243
Thus, the seventh term is −
7
243
Geometric Series
the sum of the terms in a geometric sequence given
by 𝑠𝑛 =
𝑎1 (1 −𝑟 𝑛 )
1−𝑟
Where:
𝑠𝑛 is the sum of the terms
𝑎1 is the first term
𝑟 is the common ratio
𝑛 is the number of terms
Example 1:
What is the sum of the first 8th terms of the geometric
sequence: 1, 3, 9, 27, …
Using the formula, we have:
𝑎1 (1 − 𝑟 𝑛 )
𝑠𝑛 =
1 − 𝑟
1(1 − 38 )
𝑠8 =
1 −3
𝑠8 =
𝑠8 =
𝑠8 =
1 1−6561
1−3
1 −6560
−2
−6560
−2
𝑠8 = 3280
Thus, the sum of the 8th terms is 3280.
Geometric Mean
The geometric mean of non-negative numbers a and b
is the square root of their product denoted by
𝒂𝒃
Example 1:
Insert a geometric mean between 2 and 18.
Solution:
Let x be the geometric mean
x=
(2)(18)
x = 36
x=6
ASSESSMENT:
I – Find the common difference and the nth term of each sequence.
1. 2, 5, 8, 11, …, 𝑎20
2. 4,1, -2, -5, …, 𝑎32
II – Find the sum of the first 40 terms of the arithmetic sequence given the
following conditions:
1. 7, 12, 17, 22,..
2. 𝑎1 = 5,
d=4
III – Find the common ratio and the nth term of each geometric sequence.
a.) 2, 4, 8, …𝑎16
1
4
b.) 4, 1, , … 𝑎5
https://www.cliffnotes.com/study-guides/algebra/algebraii/sequences-and-series/definition-and-examples-ofsequences
Advance with Math 10 - Ma. Edilyn Dimapilis-Chiao
Francis Joseph H. Campeña, Ph.D
Esmeralda B. Villafuerte
Authors