Number Sequences

In everyday life, we often encounter objects that are convenient to number sequentially. For example, months and quarters of the year, days of the week, building entrances and apartments, and train carriages all have numbers. Objects that are numbered with consecutive natural numbers 1, 2, 3, …, $n$ , … form a sequence.

Definition of a Sequence

📐Definition — Sequence

A sequence is a collection of objects numbered with consecutive natural numbers $1, 2, 3, \ldots, n, \ldots$

The objects that form a sequence are called the terms of the sequence. Each term of a sequence has its own number. If a term of the sequence has number $n$ , it is called the $n$ -th term of the sequence.

If the terms of a sequence are numbers, the sequence is called a numerical sequence.

Here are some examples of numerical sequences:

$1, 2, 3, 4, 5, \ldots$ — the sequence of natural numbers;
$2, 4, 6, 8, 10, \ldots$ — the sequence of even numbers;
$0.3;\; 0.33;\; 0.333;\; \ldots$ — the sequence of decimal approximations of $\dfrac{1}{3}$ ;
$19, 38, 57, 76, 95$ — the sequence of two-digit multiples of 19.

Finite and Infinite Sequences

📐Definition — Finite and Infinite Sequences

Sequences can be finite or infinite. For example, the sequence of even natural numbers is an infinite sequence, while the sequence of two-digit multiples of 19 is a finite sequence.

The terms of a sequence are denoted by letters with subscripts:

a_1, a_2, a_3, \ldots, a_n, \ldots

The subscript indicates the position number of the term. The sequence itself is denoted by $(a_n)$ , $(b_n)$ , $(c_n)$ , etc.

Ways to Define a Sequence

📐Definition — Defined Sequence

A sequence is considered defined if a rule is given that allows one to find any of its terms.

There are several main ways to define a sequence:

1. Verbal (Descriptive) Method

If the rule is described in words, this method is called descriptive. For example: “Each term of the sequence equals the remainder when its position number is divided by 3.” The first few terms are: $1, 2, 0, 1, 2, 0, 1, \ldots$

2. Definition by Table

If a sequence is finite, it can be defined using a table. For example, the table of cubes of single-digit natural numbers:

$n$	1	2	3	4	5	6	7	8	9
$a_n$	1	8	27	64	125	216	343	512	729

3. Formula for the $n$ -th Term

A sequence can be defined by a formula for the $n$ -th term. For example, the formula $x_n = 2^n$ defines the sequence of natural powers of 2:

2, 4, 8, 16, 32, \ldots

The formula $a_n = 2n - 1$ defines the sequence of odd natural numbers:

1, 3, 5, 7, 9, \ldots

4. Recurrent Method

📐Definition — Recurrent Formula

A formula that expresses a term of a sequence through one or more of its preceding terms is called a recurrent formula (from Latin recurro — to return). The conditions that determine the first or several initial terms are called initial conditions.

The method of defining a sequence using initial conditions and a recurrent formula is called the recurrent method.

For example, consider the sequence $(a_n)$ described as: the first term equals 1, and each subsequent term is 3 times the previous one:

1, 3, 9, 27, 81, \ldots

This same sequence can be defined recurrently:

a_1 = 1, \quad a_{n+1} = 3a_n \text{ for any } n \in \mathbb{N}.

Stationary Sequence

📐Definition — Stationary Sequence

A sequence in which all terms are equal is called a stationary sequence. For example, the formula $c_n = 7$ defines a stationary sequence $7, 7, 7, 7, 7, \ldots$

Connection to Functions

Consider a function $y = f(x)$ whose domain is the set of natural numbers or the set of the first $n$ natural numbers. Then the function $f$ defines an infinite sequence $f(1), f(2), \ldots, f(n), \ldots$ or a finite sequence $f(1), f(2), \ldots, f(n)$ .

In other words, an infinite sequence is a mapping from $\mathbb{N}$ to some non-empty set $A$ , and a finite sequence is a mapping from $\{1, 2, \ldots, n\}$ to some non-empty set $B$ .

For example, the function $y = x^2$ , $D(y) = \mathbb{N}$ , can be viewed as the sequence of squares of natural numbers: $1, 4, 9, 16, 25, \ldots$

Examples

✎Example — Is a Number a Term of the Sequence?

Problem. The sequence $(c_n)$ is defined by the formula $c_n = 37 - 3n$ . Is the following number a term of this sequence: 1) 19; 2) $-7$ ?

Solution.

If 19 is a term of this sequence, then there exists a natural number $n$ such that $37 - 3n = 19$ . This gives $3n = 18$ ; $n = 6$ . Therefore, 19 is the sixth term of the sequence $(c_n)$ .
We have: $37 - 3n = -7$ ; $3n = 44$ ; $n = 14\tfrac{2}{3}$ . Since $14\tfrac{2}{3}$ is not a natural number, $-7$ is not a term of this sequence.

Answer: 1) Yes, $n = 6$ ; 2) no.

✎Example — Recurrent Sequence and Divisibility

Problem. The sequence $(a_n)$ is defined recurrently: $a_1 = 15$ , $a_{n+1} = 7a_n + 1$ . Can 1001 be a term of this sequence?

Solution. Every term of the sequence $(a_n)$ is an integer that gives a remainder of 1 when divided by 7. Since 1001 is divisible by 7 ( $1001 = 7 \times 143$ ), it cannot be a term of this sequence.

Answer: no.

✎Example — Converting from n-th Term Formula to Recurrent

Problem. The sequence $(a_n)$ is defined by the formula $a_n = \dfrac{n+1}{n}$ . Express it recurrently.

Solution. We have: $a_1 = 2$ . From the formula for the $n$ -th term, express $n$ in terms of $a_n$ . We have: $na_n = n + 1$ ; $n(a_n - 1) = 1$ . Since $a_n \neq 1$ , we can write: $n = \dfrac{1}{a_n - 1}$ .

Then:

a_{n+1} = \frac{n+2}{n+1} = \frac{\dfrac{1}{a_n - 1} + 2}{\dfrac{1}{a_n - 1} + 1} = \frac{2a_n - 1}{a_n}.

Answer: $a_1 = 2$ , $a_{n+1} = \dfrac{2a_n - 1}{a_n}$ .

The Fibonacci Sequence

ℹNote — Fibonacci Numbers and the Golden Ratio

Consider the sequence $(u_n)$ defined recurrently by:

u_1 = u_2 = 1, \quad u_{n+2} = u_{n+1} + u_n.

Its first few terms are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots

The terms of this sequence are called Fibonacci numbers. The name comes from the Italian mathematician Leonardo of Pisa (Fibonacci), who, while solving a popular 12th-century problem about the number of offspring of a pair of rabbits, was the first to notice the remarkable properties of this sequence.

Fibonacci numbers have many interesting properties. If we compute the ratio $\dfrac{u_{n+1}}{u_n}$ for each $n \in \mathbb{N}$ , we get the sequence: $1; 2; 1.5; 1.(6); 1.6; 1.625; \ldots$ , which as $n$ grows approaches the number

\varphi = \frac{\sqrt{5}+1}{2} \approx 1.618.

This number is called the golden ratio. Since ancient times, people have associated this number with beauty and harmony. The ratio of the length of the Parthenon to its height is approximately 1.618.

The French mathematician Jacques Binet (1786—1856) gave a formula for the $n$ -th Fibonacci number:

u_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right].

Exercises

✏Exercise — Sequence Problems

30.2. Find the first four terms of the sequence $(a_n)$ defined by the formula:

$a_n = 4n - 3$
$a_n = \dfrac{n}{n^2 + 1}$
$a_n = \dfrac{2^n}{n}$

30.5. Find the first five terms of the sequence $(a_n)$ if:

$a_1 = 4$ , $a_{n+1} = a_n + 3$
$a_1 = -2$ , $a_2 = 6$ , $a_{n+2} = 3a_n + a_{n+1}$

30.7. The sequence $(a_n)$ is defined by the formula $a_n = 7n + 2$ . Is the following number a term of this sequence: 1) 149; 2) 47?