Sequence Concept

Numbers Assigned to Ordered Positions

A number sequence is an ordered list of numbers. Each number is called a term. We use the following notation:

The term at index $1$ is denoted by $U_1$
The term at index $2$ is denoted by $U_2$
The term at index $3$ is denoted by $U_3$
The general term is denoted by $U_n$

A sequence rule determines its later terms and may also give a formula for the general term.

More precisely, a sequence assigns one value to each allowed index:

U:\{1,2,3,\ldots\}\to\mathbb{R},\qquad n\mapsto U_n

Order is part of the definition. The sequences $1,2,3,\ldots$ and $3,2,1,\ldots$ may contain some of the same numbers, but they are not the same sequence.

Explicit and Recursive Rules

An explicit rule calculates a term directly from its index. For example:

U_n=2n+1

This gives $3,5,7,9,\ldots$ . A recursive rule states a starting term and explains how to obtain the next term:

U_1=3

Both rules describe the same sequence. The explicit form calculates any term directly from its index. The recursive form calculates each new term from the previous one.

Number Pattern from Tables and Chairs

A row of square tables and chairs gives a simple number pattern:

Table and Chair Sequence Pattern

1 Table & 4 Chair

When there is $1 \text{ table}$ with a square top, $4$ chairs can be placed around it.

If $2 \text{ tables}$ are joined together, then $6$ chairs can be placed around the combined tables.

We can create a table to observe the pattern:

Number of tables	$1$	$2$	$3$	$4$	$5$	$6$
Number of chairs	$4$	$6$	$8$	$10$	$12$	$14$

In a straight row, each table provides two chairs along the long sides. Two more chairs sit at the ends of the row. Therefore, $n$ tables give $2n$ chairs along the sides and $2$ at the ends.

Each additional $1 \text{ table}$ adds $2$ chairs. The relationship between the number of tables and chairs follows the formula:

U_n = 2n + 2

Here:

$U_n$ is the number of chairs
$n$ is the number of tables

The same pattern can be written recursively:

U_1=4

Using the Rule to Answer a Seating Question

A sequence rule can answer a counting question directly. In the example below, the rule converts a number of people into a number of tables.

Determining the Number of Chairs and Tables

The sequence rule answers questions such as:

If $20 \text{ people}$ each need a chair, how many tables must be joined?

We can use the formula $U_n = 2n + 2$ where $U_n = 20$ , so:

20 = 2n + 2

$9 \text{ tables}$ need to be joined to accommodate $20 \text{ people}$ .

Types of Sequences

Based on their patterns, number sequences fall into several types. The two that turn up most often add a fixed difference or multiply by a fixed ratio, and each of them has its own formulas.

Arithmetic Sequence

A number sequence where the difference between two consecutive terms is always constant. This difference is called the common difference ( $b$ ).

U_{n+1}=U_n+b

Example: $2, 4, 6, 8, 10, \ldots$ Common difference $(b) = 2$

Geometric Sequence

A number sequence where the ratio between two consecutive terms is always constant. This ratio is called the common ratio ( $r$ ).

U_{n+1}=rU_n

Example: $2, 6, 18, 54, \ldots$ Common ratio $(r) = 3$

Other Sequences

Besides arithmetic and geometric sequences, there are many other types of sequences such as Fibonacci sequences, quadratic sequences, cubic sequences, and more.

Example of a Fibonacci sequence: $0, 1, 1, 2, 3, 5, 8, 13, \ldots$

For indices $n\geq1$ , the sum of the two preceding terms gives the Fibonacci recursion:

F_1=0,\quad F_2=1,\quad F_{n+2}=F_{n+1}+F_n

A Finite Prefix Does Not Determine One Unique Rule

A short list of terms can suggest a plausible pattern, but it does not logically force one continuation. The prefix $1,2,4$ might continue as $8,16,\ldots$ if every term is doubled. A different rule could agree with the first three terms and change from the fourth term onward.

Two rules that match at the first three positions are $A_n=2^{n-1}$ and $B_n=1+\frac{n(n-1)}2$ . They first differ at the fourth position:

Position	Doubling rule	Quadratic rule
$1$	$1$	$1$
$2$	$2$	$2$
$3$	$4$	$4$
$4$	$8$	$7$
$5$	$16$	$11$

A fully defined sequence needs an unambiguous rule or enough conditions to identify it. Guessing a pattern produces a conjecture. A proof requires a rule that determines every term.

Finding Patterns

To determine the pattern of a sequence:

Observe the differences between consecutive terms
Check if the difference is constant (arithmetic sequence)
If not, check if the ratio is constant (geometric sequence)
If neither, check for other possible patterns

A sequence assigns one term to each index. Its rule may be explicit, recursive, or defined in another way.

Practice

Consider the sequence $V_n=n^2-n$ for positive integers $n$ . The explicit formula calculates a term from its position. A recursive rule needs an initial value and a relationship between consecutive terms.

Find the first five terms.
Give a recursive rule for the sequence.

Worked Solutions

Problem 1. Substitute $n=1,2,3,4,5$ into the given term formula.

V_1,V_2,V_3,V_4,V_5=0,2,6,12,20

Problem 2. A recursive rule expresses the next term using the previous term:

V_1=0,\qquad V_{n+1}=V_n+2n

The recursive increment is $2n$ because:

\begin{aligned} V_{n+1}-V_n &=((n+1)^2-(n+1))-(n^2-n) \\ &=n^2+2n+1-n-1-n^2+n \\ &=2n \end{aligned}