THE numerical sequence, as the name suggests, is a sequence of numbers and usually has a recurrence law, which makes it possible to predict what the next terms will be getting to know your predecessors. We can assemble number sequences with different criteria, such as a sequence of even numbers, or sequence of numbers divisible by 4, sequence of prime numbers, sequence of perfect squares, finally, there are several possibilities of sequences numerical.
When we rank the sequence in terms of the number of terms, the sequence can be finite or infinite. When we classify the sequence in terms of the behavior of the terms, this sequence can be ascending, descending, oscillating or constant. There are special cases of sequences that are known as arithmetic progressions and geometric progressions.
Read too: How to calculate soma of the terms of a arithmetic progression?
Number sequence summary
The numerical sequence is nothing more than a sequence of numbers.
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Some numerical sequence examples:
sequence of even numbers (0,2,4,6,8…);
sequence of naturals less than 6 (1, 2, 3, 4, 5);
sequence of prime numbers (2,3,5,7,11,…).
The law of formation of a progression is the rule that governs this sequence.
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A sequence can be finite or infinite.
Finite: when you have a limited amount of terms.
Infinite: when you have an unlimited amount of terms.
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A sequence can be increasing, disbelieving, constant or fluctuating.
Crescent: when the term is always smaller than its successor.
Descending: when the term is always greater than its successor.
Constant: when the term is always equal to its successor.
Oscillating: when there are terms larger and smaller than its successor.
There are special cases of sequence known as arithmetic progression or geometric progression.
Law of occurrence of number sequence
We know as numerical sequence any sequence formed by numbers. We usually demonstrate sequences by listing their terms, enclosed in parentheses and separated by a comma. This list is known as the law of occurrence of a number sequence.
(The1, a2, a3, …, ano)
The1 → 1st term of the sequence
The2 → 2nd term of the sequence
The3 → 3rd term of the sequence
Theno → nth term of the sequence
Let's look at some examples below.
Example 1:
Law of occurrence of sequence of numbers multiples of 5:
(0, 5, 10, 15, 20, 25, …)
Example 2:
Law of occurrence of the sequence of Prime numbers:
(2,3,5,7,11,13,17,19,23 … )
Example 3:
Law of occurrence of whole negative:
( – 1, – 2, – 3, – 4, – 5, – 6, – 7...)
Example 4:
Sequence of odd numbers less than 10:
(1, 3, 5, 7, 9)
Read too: What are the properties of odd and even numbers?
Numerical Sequence Classification
There are two distinct ways to classify a string. The first one is as to the amount of terms, the way in which a sequence can be finite or infinite. The other way to classify sequences is as to their behavior. In this case, they are classified as increasing, decreasing, constant or fluctuating.
Classification by the amount of terms
→ finite number sequence
The sequence is finite when it has a limited amount of terms.
Examples:
(1, 2, 3, 4, 5)
(– 16, – 8, – 4, – 2, – 1)
→ infinite number sequence
The sequence is infinite when it has an unlimited amount of terms.
Examples:
(10, 100, 1.000, 10.000, 100.000, 1.000.000 … )
(– 5, – 8, – 11, – 14, – 17, – 20, – 23 … )
Behavior rating
→ Ascending number sequence
A sequence is ascending when any term is always smaller than its successor in sequence.
Examples:
(0, 1, 2, 3, 4, 5, 6, … )
( – 5, – 3, – 1, 1, 3, 5, 7)
→ Descending number sequence
A sequence is descending when any term is always greater than its successor in sequence.
Examples:
(10, 7, 4, 1, – 2, – 5, – 8 … )
(4, – 8, – 16, – 32, – 64 )
→ constant number sequence
A sequence is constant when all terms in the sequence are the same:
Examples:
(1, 1, 1, 1, 1, 1, 1,)
( – 4, – 4, – 4, – 4 … )
→ Oscillating Number Sequence
A sequence is swinging when there are terms that are bigger and terms that are smaller that their respective successors in the sequence:
Examples:
(1,-2,4,-8,16,-32,64...)
(1, – 1, 1, – 1, 1, – 1)
Number Sequence Formation Law
Some sequences can be described by a formula that generates your terms. This formula is known as the law of formation. We use the law of formation to find any term in the sequence when we know its behavior.
Example 1:
The following sequence is formed by perfect squares:
(0, 1, 4, 9, 16, 25, 36, 64, … )
We can describe this sequence by the law of formation:
Theno = (n – 1)²
n → term number
Theno → the position term no
With this formula, it is possible to know, for example, the term that occupies position number 10 in the sequence:
The10 = ( 10 – 1) ²
The10 = 9²
The10 = 81
Example 2:
List the terms of the sequence whose formation law is theno = 2n – 5.
To list, we'll find the first terms in the sequence:
1st term:
Theno = 2n - 5
The1 = 2·1 – 5
The1 = 2 – 5
The1 = – 3
2nd term:
Theno = 2n - 5
The2 = 2·2 – 5
The2 = 4 – 5
The2 = – 1
3rd term:
Theno = 2n - 5
The3 = 2·3 – 5
The3 = 6 – 5
The3 = 1
4th term:
Theno = 2n - 5
The4 = 2·4 – 5
The4 = 8 – 5
The4 = 3
5th term:
The5 = 2n - 5
The5 = 2·5 – 5
The5 = 10 – 5
The5 = 5
So the sequence is:
(– 1, 1, 3, 5 … )
See too: Roman numbers — numerical system that uses letters to represent values and quantities
Arithmetic progression and geometric progression
They exist special cases of sequences which are known as arithmetic progression and geometric progression. A sequence is a progression when there is a reason for a term for its successor.
arithmetic progression
When we know the first term in the sequence and, to find the second,we add the first to a value r and to find the third term, we add the second to this same value. r, and so on, the string is classified as a arithmetic progression.
Example:
(1, 5, 9, 13, 17, 21, …)
This is an arithmetic progression of ratio equal to 4 and first term equal to 1.
Note that to find the successor of a number in the sequence, just add 4, so we say that 4 is the reason for this arithmetic progression.
Geometric progression
At geometric progression, there is also a reason, but in this case, to find the successor of a term, we must multiply the term by the ratio.
Example:
(2, 6, 18, 54, 162, … )
This is a geometric progression of ratio equal to 3 and first term equal to 2.
Note that to find the successor of a number in this sequence, simply multiply by 3, which makes the ratio of this geometric progression to be 3.
solved exercisesabout number sequence
Question 1 - Analyzing the sequence (1, 4, 9, 16, 25, … ), we can say that the next two numbers will be:
A) 35 and 46.
B) 36 and 49.
C) 30 and 41.
D) 41 and 66.
Resolution
Alternative B.
To find the terms of the sequence, it is important to find a regularity in the sequence, that is, to understand its law of occurrence. Note that, from the first term to the second term, we add 3; from the second to the third term, we add 5; from the third to the fourth term and from the fourth to the fifth term, we add, respectively, 7 and 9, so the sum increases by two units to each term of the sequence, that is, in the next, we will add 11, then 13, then 15, then 17 and so on successively. To find 25's successor, we'll add 11.
25 + 11 = 36.
To find the successor of 36, we'll add 13.
36 + 13 = 49
So the next terms will be 36 and 49.
Question 2 - (AOCP Institute) Next, a numerical sequence is presented, such that the elements of this sequence were arranged obeying a (logic) law of formation, where x and y are whole numbers: (24, 13, 22, 11, 20, 9, x, y). Observing this sequence and finding the values of x and y, following the law of formation of the given sequence, it is correct to state that
A) x is a number greater than 30.
B) y is a number less than 5.
C) the sum of x and y results in 25.
D) the product of x and y gives 106.
E) the difference between y and x, in that order, is a positive number.
Resolution
Alternative C.
We want to find the 7th and 8th term of this sequence.
Analyzing the law of occurrence of the sequence (24, 13, 22, 11, 20, 9, x, y), it is possible to see that there is a logic for the odd terms (1st term, 3rd term, 5th term … ). Note that the 3rd term is equal to the 1st term minus 2, since 24 – 2 = 22. Using this same logic, the 7th term, represented by x, will be the 5th term minus 2, that is, x = 20 – 2 = 18.
There is a similar logic for the even terms (2nd term, 4th term, 6th term…): the 4th term is the 2nd term minus 2, since 13 – 2 = 11, and so on. We want the 8th term, represented by y, which will be the 6th term minus 2, so y = 9 – 2 = 7.
So we have x = 18 and y = 7. Analyzing the alternatives, we have that x + y = 25, that is, the sum of x and y results in 25.
By Raul Rodrigues de Oliveira
Maths teacher
Source: Brazil School - https://brasilescola.uol.com.br/matematica/sequencia-numerica.htm