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Chapter 6: Sequence and Series
Arithmetic Sequence:
A sequence of terms in which the difference
between any two consecutive terms is
constant.
Example:
5, 8, 11, 14, 17, 20, 23,….
The common difference is 3
Geometric Sequence:
A sequence of terms in which the ratio
between any two consecutive terms is
constant.
Example:
2, 4, 8, 16, 32, 62,…
The common ratio is 2
Formulas for Arithmetic Sequences:
Explicit:
𝒂𝒏 = 𝒂𝟏 + 𝒅(𝒏 − 𝟏)
𝑎1 is the first term
d is the common difference
n is the nth term in the sequence
Formulas for Geometric Sequences:
Explicit:
𝒂𝒏 = 𝒂𝟏 (𝒓𝒏−𝟏 )
𝑎1 is the first term
r is the common ratio
n is the nth term in the sequence
example:
Find the 40 term of the following sequence:
5, 8, 11, 14, 17….
example:
Find the 30 term of the following sequence:
2, 4, 8, 16…
d=3
first term = 5
n = 40
𝑎𝑛 = 5 + 3(40 − 1)
𝑎𝑛 = 5 + 3(39)
𝑎𝑛 = 5 + 117
𝑎𝑛 = 122
First term = 2
r=2
n = 30
Recursive:
Recursive:
𝒂𝒏 = 𝒂𝒏−𝟏 + 𝒅
𝑎𝑛−1 is the previous term to 𝑎𝑛
d is the common difference
𝒂𝒏 = 𝒂𝒏−𝟏 (𝒓)
𝑎𝑛−1 is the previous term to 𝑎𝑛
r is the common ratio
𝑎𝑛 = 2(230−1 )
𝑎𝑛 = 2(536870912)
𝑎𝑛 = 1073741824
Series:
The sum of terms in a given sequence. Can be Arithmetic or Geometric.
The sum of the first n terms of a series is denoted as 𝑆𝑛 .
Series can be finite (have a final term) or infinite.
Finite Series:
A series with a finite number of terms.
Infinite Series:
A series with an infinite number of terms.
An infinite Series can have a finite Sum
(Convergent Series) or not have a Sum
(Divergent Series).
Sum formula for a finite Arithmetic Series:
𝒂𝟏 + 𝒂𝒏
)
𝟐
n is the number of terms
𝑎1 is the first term
𝑎𝑛 is the nth term
Sum of an Infinite Geometric Series:
𝒂𝟏
𝑺=
𝟏−𝒓
Only possible when |𝑟| < 1. Diverges if not.
𝑺𝒏 = 𝒏(
Sum formula for a finite Geometric Series:
𝟏 − 𝒓𝒏
𝑺𝒏 = 𝒂𝟏 (
)
𝟏−𝒓
Example:
Find the sum for the infinite geometric series
if possible.
5 + 4 + 3.2 + 2.56+. ..
r = .8 so it will converge
𝑆=
Summation Notation:
5
5
= = 25
1 − .8 . 2
𝒏
∑ 𝒂𝒌
𝒌=𝟏
𝑘 = 1 index of summation (what term you begin on)
𝑛 upper bound of summation (what term you end on)
𝑎𝑘 an indexed variable representing each successive term in the series
Example:
4
∑(2𝑘 − 1) = (2(𝟏) − 1) + (2(𝟐) − 1) + (2(𝟑) − 1) + (2(𝟒) − 1) = 16
𝑘=1
Geometric Mean:
The square root of the product of two
numbers (used to find the middle term in a
geometric sequence).
√𝒂𝒃
Example:
Find the missing value in the geometric
sequence: 16, ____ , 25.
√16 ∗ 25
√400
20
Arithmetic Mean:
The average of a set of numerical values,
calculated by adding the values together and
dividing by the number of terms in the set
(used to find the middle term in an arithmetic
sequence).