A numerical sequence is a set of numbers organized in an orderly manner. The numerical sequence can be assembled using different criteria — for example, the sequence of even numbers or the sequence of multiples of 3. When we can describe this criterion with a formula, we call this formula the law of formation of the numerical sequence.
Read too: Differences between number, numeral and digit
Summary about numerical sequence
Number sequence is a list of numbers arranged in order.
The numerical sequence can follow different criteria.
The law of occurrence of the numerical sequence is the list of elements that exist in the sequence.
The sequence can be classified in two ways. One takes into account the number of elements, and the other takes into account behavior.
As for the number of elements, the sequence can be finite or infinite.
As for behavior, the sequence can be increasing, constant, decreasing or oscillating.
When the numerical sequence can be described by an equation, this equation is known as the law of formation of the numerical sequence.
What are sequences?
The sequences are sets of elements arranged in a certain order. In our daily lives, we can perceive several situations that involve sequences:
Sequence of months: January, February, March, April,..., December.
Sequence of years of the first 5 World Cups of the 21st century: 2002, 2006, 2010, 2014, 2018.
There are several other possible sequences, such as name sequence or age sequence. Whenever there is an established order, there is a sequence.
Each element of a sequence is known as a term of the sequence, so in a sequence there is the first term, second term and so on. Generally, a sequence can be represented by:
\((a_1,a_2,a_3,…,a_n )\)
\(to 1\) → the first term.
\(a_2\) → the second term.
\(a_3\) → the third term.
\(a_n\) → any term.
Law of occurrence of the numerical sequence
We can have sequences of various elements, such as months, names, days of the week, among others. Asequence is a numerical sequence when it involves numbers. We can form the sequence of even numbers, odd numbers, Prime numbers, multiples of 5 etc.
The sequence is represented using an occurrence law. The law of occurrence is nothing more than the list of elements of the numerical sequence.
Examples:
(1, 3, 5, 7, 9, 11, 13, 15) → sequence of odd numbers from 1 to 15.
(0, 5, 10, 15, 20, 25, 30, ...) → sequence of numbers that are multiples of 5.
(-1, 1, -1, 1, -1, 1) → alternating sequence between 1 and -1.
What is the classification of the numerical sequence?
We can classify sequences in two different ways. One of them is taking into account the number of elements, and the other is taking into account the behavior of these elements.
→ Classification of the numerical sequence according to the number of elements
When we classify the sequence according to the number of elements, there are two possible classifications: the finite sequence and the infinite sequence.
◦ Finite number sequence
A sequence is finite if it has a limited number of elements.
Examples:
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
(0, 0, 0, 0, 0, 0, 0, 0, 0)
(-4, -6, -8, -10, -12)
◦ Infinite number sequence
A sequence is infinite if it has an unlimited number of elements.
Examples:
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...)
(3, 0, -3, -6, -9, -12, ...)
( -1, 2, -4, 8, -16, ...)
→ Classification of the numerical sequence according to the behavior of the sequence
The other way to classify is by sequence behavior. In this case, the sequence can be increasing, constant, oscillating or decreasing.
◦ Increasing number sequence
The sequence is increasing if a term is always greater than its predecessor.
Examples:
(1, 5, 9, 13, 17, ...)
(10, 11, 12, 13, 14, 15, ...)
◦ Constant number sequence
The sequence is constant when all terms have the same value.
Examples:
(1, 1, 1, 1, 1, 1, 1, ...)
(-1, -1, -1, -1, -1, ...)
◦ Descending number sequence
The sequence is decreasing if the terms in the sequence are always smaller than their predecessors.
Examples:
(-1, -2, -3, -4, -5, ...)
(19, 16, 13, 10, 8, ...)
◦ Oscillating number sequence
The sequence is oscillating if there are terms greater than their predecessors and terms smaller than their predecessors alternately.
Examples:
(1, -3, 9, -27, 81, ...)
(1, -1, 2, -2, 3, -3, 4, -4, ...)
Law of formation of the numerical sequence
In some cases, it is possible to describe the sequence using a formula, however this is not always possible. For example, the sequence of prime numbers is a well-defined sequence, however we cannot describe it using a formula. Knowing the formula, we were able to construct the law of occurrence of the numerical sequence.
Example 1:
Sequence of even numbers greater than zero.
\(a_n=2n\)
Note that when replacing n for one natural number (1, 2, 3, 4, ...), we will find an even number:
\(a_1=2⋅1=2\)
\(a_2=2⋅2=4\)
\(a_3=2⋅3=6\)
\(a_4=2⋅4=8\)
So, we have a formula that generates the terms of the sequence formed by even numbers greater than zero:
(2, 4, 6, 8, ...)
Example 2:
Sequence of natural numbers greater than 4.
\(a_n=4+n\)
Calculating the terms of the sequence, we have:
\(a_1=4+1=5\)
\(a_2=4+2=6\)
\(a_3=4+3=7\)
\(a_4=4+4=8\)
Writing the law of occurrence:
(5, 6, 7, 8,…)
See too: Arithmetic progression — a special case of numerical sequence
Solved exercises on numerical sequence
Question 1
A numerical sequence has a formation law equal to \(a_n=n^2+1\). Analyzing this sequence, we can state that the value of the 5th term of the sequence will be:
A) 6
B) 10
C) 11
D) 25
E) 26
Resolution:
Alternative E
Calculating the value of the 5th term of the sequence, we have:
\(a_5=5^2+1\)
\(a_5=25+1\)
\(a_5=26\)
Question 2
Analyze the following numerical sequences:
I. (1, -2, 3, -4, 5, -6, ...)
II. (13, 13, 13, 13, 13, ...)
III. (1, 2, 3, 4, 5, 6, ...)
We can state that sequences I, II and III are classified respectively as:
A) increasing, oscillating and decreasing.
B) decreasing, increasing and oscillating.
C) oscillating, constant and increasing.
D) decreasing, oscillating and constant.
E) oscillating, decreasing and increasing.
Resolution:
Alternative C
Analyzing the sequences, we can state that:
I. (1, -2, 3, -4, 5, -6, ...)
It is oscillating, as there are terms that are greater than their predecessors and terms that are smaller than their predecessors.
II. (13, 13, 13, 13, 13, ...)
It is constant, as the terms of the sequence are always the same.
III. (1, 2, 3, 4, 5, 6, ...)
It is increasing, as the terms are always larger than their predecessors.
