4.1 Sequences and Their Terms | Sequence And Series | Mathematics 11

A sequence is a function whose domain is the set of positive integers ${1, 2, 3, \dots}$ . Each value $a_{n}$ is called the $n^{t h}$ term or general term of the sequence.

$a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots$

Finding Terms from a General Formula

To find the $k^{t h}$ term, substitute $n = k$ into the formula $a_{n}$ .

Example: If $a_{n} = \frac{n ^{2} - 1}{n ^{2} + 1}$ , find $a_{1}, a_{2}, a_{3}$ .

$a_{1} = \frac{1 - 1}{1 + 1} = 0, a_{2} = \frac{4 - 1}{4 + 1} = \frac{3}{5}, a_{3} = \frac{9 - 1}{9 + 1} = \frac{8}{10} = \frac{4}{5}$

Arithmetic Sequences

A sequence $a_{1}, a_{2}, a_{3}, \dots$ is arithmetic if there is a constant common difference $d$ such that:

$a_{n + 1} - a_{n} = d for all n$

The general term of an arithmetic sequence is:

$a_{n} = a_{1} + (n - 1) d$

Example: The sequence $5, 8, 11, 14, \dots$ has $a_{1} = 5$ and $d = 3$ . $a_{n} = 5 + (n - 1) (3) = 3 n + 2$

Geometric Sequences

A sequence is geometric if there is a constant common ratio $r$ such that:

$\frac{a _{n + 1}}{a _{n}} = r for all n$

The general term of a geometric sequence is:

$a_{n} = a_{1} \cdot r^{n - 1}$

Example: The sequence $3, 9, 27, 81, \dots$ has $a_{1} = 3$ and $r = 3$ . $a_{n} = 3 \cdot 3^{n - 1} = 3^{n}$

Identifying Arithmetic vs Geometric Sequences

Property	Arithmetic	Geometric
Test	$a_{n + 1} - a_{n} = const$	$\frac{a _{n + 1}}{a _{n}} = const$
General term	$a_{1} + (n - 1) d$	$a_{1} \cdot r^{n - 1}$
Example	$2, 5, 8, 11, \dots$	$2, 6, 18, 54, \dots$

Alternating Sequences

An alternating sequence changes sign between consecutive terms. This is achieved using $(- 1)^{n}$ or $(- 1)^{n + 1}$ :

$(- 1)^{n}$ : first term is negative (since $(- 1)^{1} = - 1$ )
$(- 1)^{n + 1}$ : first term is positive (since $(- 1)^{2} = + 1$ )

Example: $a_{n} = (- 1)^{n + 1} \cdot n$ gives $1, - 2, 3, - 4, 5, \dots$

Finding the General Term from a Pattern

When given the first few terms, identify the pattern and express it as a function of $n$ .

Example 1: $1 \cdot 2, 2 \cdot 3, 3 \cdot 4, \dots$

First factor: $1, 2, 3, \dots = n$
Second factor: $2, 3, 4, \dots = n + 1$
General term: $a_{n} = n (n + 1)$

Example 2: $2, 4, 6, 8, \dots$

Numbers under root: $2, 4, 6, 8, \dots = 2 n$
General term: $a_{n} = 2 n$

Property

Arithmetic

Geometric

Test

a_{n + 1} - a_{n} = const

\frac{a _{n + 1}}{a _{n}} = const

General term

a_{1} + (n - 1) d

a_{1} \cdot r^{n - 1}

Example

2, 5, 8, 11, \dots

2, 6, 18, 54, \dots

Finding the General Term from a Pattern

When given the first few terms, identify the pattern and express it as a function of

n

Example 1:

1 \cdot 2, 2 \cdot 3, 3 \cdot 4, \dots

First factor:

1, 2, 3, \dots = n

Second factor:

2, 3, 4, \dots = n + 1

General term:

a_{n} = n (n + 1)

Example 2:

2, 4, 6, 8, \dots

Numbers under root:

2, 4, 6, 8, \dots = 2 n

General term:

a_{n} = 2 n