Numerical sequences and ways to set them. Task for practical work "Specifying numerical sequences in various ways, calculating members of a sequence

In this lesson, we will begin the study of progressions. Here we will get acquainted with the number sequence and how to set it.

First, we recall the definition and properties of functions of numerical arguments and consider a special case of a function when x belongs to the set natural numbers. We give a definition of a numerical sequence and give some examples. We will show an analytical way of specifying a sequence through the formula of its n-th member and consider several examples for specifying and determining a sequence. Next, consider the verbal and recurrent assignment of a sequence.

Theme: Progressions

Lesson: Numeric sequence and how to set it

1. Repetition

Numeric sequence, as we will see, this is a special case of a function, so let's remember the function definition.

A function is a law according to which each valid value of an argument is assigned a unique value of the function.

Here are examples of known functions.

Rice. 1. Graph of a function

All values ​​are allowed except 0. The graph of this function is a hyperbola (see Fig.1).

2.. All values ​​are allowed, .

Rice. 2. Function Graph

Schedule quadratic function- parabola, characteristic points are also marked (see Fig. 2).

3..

Rice. 3. Graph of a function

All x values ​​are allowed. The graph of a linear function is a straight line (see Fig. 3).

2. Definition of a numerical sequence

If x takes only natural values ​​(), then we have a special case, namely a numerical sequence.

Recall that natural numbers are 1, 2, 3, …, n, …

The function , where , is called a function of a natural argument, or a numerical sequence, and is denoted as follows: or , or .

Let us explain what, for example, the notation means.

This is the value of the function when n=1, i.e. .

This is the value of the function when n=2 i.e. etc...

This is the value of the function when the argument is n, i.e. .

3. Sample sequences

1. is the general term formula. We set different values ​​of n, we get different values ​​of y - members of the sequence.

When n=1; , when n=2, etc., .

Numbers are members of a given sequence, and points lie on the hyperbola - the graph of the function (see Fig. 4).

Rice. 4. Function Graph

If n=1, then ; if n=2, then ; if n=3, then etc.

The numbers are members of a given sequence, and the points lie on a parabola - the graph of a function (see Fig. 5).

Rice. 5. Function Graph

Rice. 6. Function Graph

If n=1, then ; if n=2 then ; if n=3 then etc.

Numbers are members of a given sequence, and the points lie on a straight line - the graph of the function (see Fig. 6).

4. Analytical method for specifying the sequence

There are three ways to specify sequences: analytical, verbal, and recurrent. Let's consider each of them in detail.

The sequence is given analytically if the formula of its nth term is given.

Let's look at a few examples.

1. Find several members of the sequence, which is given by the formula of the n-th member: (an analytical way of specifying the sequence).

Decision. If n=1, then ; if n=2, then ; if n=3 then etc.

For a given sequence, we find and .

.

.

2. Consider the sequence given by the formula of the nth member: (analytical way of specifying the sequence).

Let's find several members of this sequence.

If n=1, then ; if n=2 then ; if n=3 then etc.

In general, it is not difficult to understand that the members of this sequence are those numbers that, when divided by 4, give a remainder of 1.

a. For a given sequence, find .

Decision: . Answer: .

b. Two numbers are given: 821, 1282. Are these numbers members of the given sequence?

In order for the number 821 to be a member of the sequence, it is necessary that the equality: or . The last equality is an equation for n. If the decision given equation is a natural number, then the answer is yes.

In this case, it is. .

Answer: yes, 821 is a member of the given sequence, .

Let's move on to the second number. Similar reasoning leads us to the solution of the equation: .

Answer: since n is not a natural number, the number 1282 is not a member of the given sequence.

Formulas that analytically define a sequence can be very different: simple, complex, etc. The requirement for them is the same: each value of n must correspond to a single number.

3. Given: the sequence is given by the following formula.

Find the first three members of the sequence.

, , .

Answer: , , .

4. Are the numbers members of the sequence?

a. , i.e. . Solving this equation, we get that . This is a natural number.

Answer: the first given number is a member of this sequence, namely its fifth member.

b. , i.e. . Solving this equation, we get that . This is a natural number.

Answer: the second given number is also a member of this sequence, namely its ninety-ninth member.

5. Verbal way of setting the sequence

We have considered an analytical way of specifying a numerical sequence. It is convenient, common, but not the only one.

The next way is verbal assignment of the sequence.

The sequence, each of its members, the possibility of calculating each of its members can be specified in words, not necessarily formulas.

Example 1 A sequence of prime numbers.

Recall that a prime number is a natural number that has exactly two distinct divisors: 1 and the number itself. Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.

There are countless of them. Euclid also proved that the sequence of these numbers is infinite, that is, there is no largest prime number. The sequence is given, each term can be calculated, tedious, but can be calculated. This sequence is given verbally. Unfortunately the formulas are not available.

Example 2 Consider the number =1.41421…

This is irrational number, its decimal notation provides for an infinite number of digits. Let's consider a sequence of decimal approximations of a number by deficiency: 1; 1.4; 1.41; 1.414; 1.4142; etc.

There are an infinite number of members of this sequence, each of them can be calculated. It is impossible to set this sequence by a formula, so we describe it verbally.

6. Recursive way to specify a sequence

We have considered two ways of specifying a numerical sequence:

1. Analytical method, when the formula of the nth member is given.

2. Verbal assignment of the sequence.

And, finally, there is a recurrent sequencing, when the rules for calculating the nth term from the previous terms are given.

Consider

Example 1 Fibonacci sequence (13th century).

History reference:

Leonardo of Pisa (about 1170, Pisa - about 1250) - the first major mathematician medieval Europe. He is best known by the nickname Fibonacci.

Much of what he learned he set forth in his outstanding Book of the Abacus (Liber abaci, 1202; only the supplemented manuscript of 1228 has survived to this day). This book contains almost all the arithmetic and algebraic information of that time, presented with exceptional completeness and depth. The "Book of the abacus" rises sharply above the European arithmetic and algebraic literature of the 12th-14th centuries. the variety and strength of methods, the richness of tasks, the evidence of presentation. Subsequent mathematicians widely drew from it both problems and methods for solving them. According to the first book, many generations of European mathematicians studied the Indian positional number system.

The first two terms are given and each subsequent term is the sum of the previous two

one; one; 2; 3; 5; eight; thirteen; 21; 34; 55; ... are the first few members of the Fibonacci sequence.

This sequence is given recursively, nth term depends on the previous two.

Example 2

In this sequence, each subsequent term is greater than the previous one by 2. Such a sequence is called an arithmetic progression.

The numbers 1, 3, 5, 7... are the first few members of this sequence.

Let's give one more example of a recurrent assignment of a sequence.

Example 3

The sequence is given as follows:

Each subsequent term of this sequence is obtained by multiplying the previous term by the same number q. Such a sequence has a special name - a geometric progression. Arithmetic and geometric progressions will be the objects of our study in the next lessons.

Let's find some members of the specified sequence at b=2 and q=3.

Numbers 2; 6; eighteen; 54; 162 ... are the first few members of this sequence.

Interestingly, this sequence can also be specified analytically, i.e., you can choose a formula. In this case, the formula will be as follows.

Indeed: if n=1, then ; if n=2, then ; if n=3 then etc.

Thus, we state that the same sequence can be given both analytically and recurrently.

7. Summary of the lesson

So, we have considered what a numerical sequence is and how to set it.

In the next lesson, we will get acquainted with the properties of numerical sequences.

1. Makarychev Yu. N. et al. Algebra grade 9 (textbook for secondary school).-M.: Education, 1992.

2. Makarychev Yu. N., Mindyuk N. G., Neshkov, K. I. Algebra for Grade 9 with deepening. study mathematics.-M.: Mnemozina, 2003.

3. Makarychev Yu. N., Mindyuk N. G. Additional chapters to the school textbook of algebra grade 9.-M .: Education, 2002.

4. Galitsky M. L., Goldman A. M., Zvavich L. I. Collection of problems in algebra for 8-9 classes ( tutorial for students of schools and classes with deepening. study mathematics).-M.: Education, 1996.

5. Mordkovich A. G. Algebra grade 9, textbook for general education institutions. - M.: Mnemosyne, 2002.

6. Mordkovich A. G., Mishutina T. N., Tulchinskaya E. E. Algebra grade 9, problem book for educational institutions. - M.: Mnemosyne, 2002.

7. Glazer G. I. History of mathematics at school. Grades 7-8 (a guide for teachers).-M.: Enlightenment, 1983.

1. College section. ru in mathematics.

2. Portal of Natural Sciences.

3. Exponential. ru Educational mathematical site.

1. No. 331, 335, 338 (Makarychev Yu. N. et al. Algebra Grade 9).

2. No. 12.4 (Galitsky M. L., Goldman A. M., Zvavich L. I. Collection of problems in algebra for grades 8-9).

Algebra. Grade 9
Lesson #32
The date:_____________
Teacher: Gorbenko Alena Sergeevna
Topic: Numerical sequence, ways to set it and properties
Lesson type: combined
The purpose of the lesson: to give the concept and definition of a numerical sequence, to consider ways
assignments of numerical sequences
Tasks:
Educational: to familiarize students with the concept of a numerical sequence and a member
numerical sequence; familiarize yourself with analytical, verbal, recurrent and
graphical ways of setting a numerical sequence; consider the types of numbers
sequences; preparation for EAEA;
Developing: development of mathematical literacy, thinking, calculation techniques, skills
comparisons when choosing a formula; instilling an interest in mathematics;
Educational: education of skills of independent activity; clarity and
organization in work; enable every student to succeed;
Equipment: School supplies, blackboard, chalk, textbook, handouts.
During the classes
I. Organizing time
 Mutual greeting;
 Fixing absentees;
 Announcement of the topic of the lesson;
 Setting goals and objectives of the lesson by students.
Sequence is one of the most basic concepts in mathematics. The sequence can
be composed of numbers, points, functions, vectors, etc.
Today in the lesson we will get acquainted with the concept of "numerical sequence", we will find out what
there may be sequences, let's get acquainted with the famous sequences.

II. Updating of basic knowledge.
Do you know functions defined on the entire number line or on its continuous
III.
intervals:
linear function y \u003d kx + v,
quadratic function y \u003d ax2 + inx + c,


 function y =



 function y = |x|.
Preparation for the perception of new knowledge
direct proportionality y \u003d kx,
inverse proportionality y \u003d k / x,
cubic function y = x3,
,
But there are functions defined on other sets.
Example. Many families have a custom, a kind of ritual: on the birthday of a child
parents bring him to door frame and solemnly celebrate the growth of the birthday man on it.
The child grows, and over the years, a whole ladder of marks appears on the jamb. Three, five, two: This is
sequence of growth from year to year. But there is another sequence, namely
its members are carefully written out next to the serifs. This is a sequence of growth values.
The two sequences are related to each other.
The second is obtained from the first by addition.
Growth is the sum of the gains for all previous years.
Consider a few more issues.
Task 1. There are 500 tons of coal in the warehouse, 30 tons are delivered every day. How much coal will be
in stock in 1 day? 2 day? Day 3? Day 4? Day 5?
(Students' answers are written on the board: 500, 530, 560, 590, 620).
Task 2. During the period of intensive growth, a person grows by an average of 5 cm per year. Now growth
student S. is 180 cm. How tall will he be in 2026? (2m 30 cm). But this is not to be
maybe. Why?
Task 3. Every day, every person with influenza can infect 4 others.
In how many days will all the students of our school (300 people) fall ill? (After 4 days).
These are examples of functions defined on the set of natural numbers - numerical
sequences.
The goal of the lesson is: Find ways to find any member of the sequence.
Lesson objectives: Find out what a numerical sequence is and how
sequences.
IV. Learning new material
Definition: A numerical sequence is a function defined on a set
natural numbers (sequences constitute such elements of nature that
can be numbered).
The concept of a numerical sequence arose and developed long before the creation of the doctrine of
functions. Here are examples of infinite number sequences known back in
antiquities:
1, 2, 3, 4, 5, : sequence of natural numbers;
2, 4, 6, 8, 10, : sequence of even numbers;
1, 3, 5, 7, 9, : sequence of odd numbers;
1, 4, 9, 16, 25, : sequence of squares of natural numbers;
2, 3, 5, 7, 11, : sequence of prime numbers;
,
1,
The number of members of each of these series is infinite; first five sequences
, : sequence of reciprocals of natural numbers.
,
monotonically increasing, the latter monotonically decreasing.

Designation: y1, y2, y3, y4, y5,:
1, 2, 3, 4, 5, :p,:sequence number of the sequence member.
(yn) sequence, ynth member of the sequence.
(an) sequence, nth member of the sequence.
an1 is the previous member of the sequence,
an+1 subsequent member of the sequence.
Sequences are finite and infinite, increasing and decreasing.
Tasks for students: Write down the first 5 members of the sequence:
From the first natural number increase by 3.
From 10 increase by 2 times and decrease by 1.
From the number 6, alternate an increase of 2 and an increase of 2 times.
These number series are also called number sequences.
Sequencing methods:
verbal way.
Sequencing rules are described in words, without formulas or
when there are no regularities between the elements of the sequence.
Example 1. A sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, .... .
Example 2. An arbitrary set of numbers: 1, 4, 12, 25, 26, 33, 39, ... .
Example 3. Sequence of even numbers 2, 4, 6, 8, 10, 12, 14, 16, ...
analytical way.
Any nth element of the sequence can be determined using a formula.
Example 1. Sequence of even numbers: y = 2n.
Example 2. The sequence of the square of natural numbers: y = n2;
1, 4, 9, 16, 25, ..., n2, ... .
Example 3. Stationary sequence: y = C; C, C, C, ...,C, ...
special case: y=5; 5, 5, 5, ..., 5, ... .
Example 4. Sequence y = 2n;
2, 22, 23, 24, ..., 2n, ... .
recursive way.
A rule is specified that allows calculating the nth element of the sequence if
its previous elements are known.
Example 1. Arithmetic progression: a1=a, an+1=an+d, where a and d are given numbers, d
difference of an arithmetic progression. Let a1=5, d=0.7, then the arithmetic progression
will look like: 5; 5.7; 6.4; 7.1; 7.8; 8.5; ... .
Example 2. Geometric progression: b1= b, bn+1= bnq, where b and q are given numbers, b
0,
0; q is the denominator geometric progression. Let b1=23, q=½, then the geometric
q
the progression will look like: 23; 11.5; 5.75; 2.875; ... .
4) Graphical way. Numeric sequence
given by a graph which is
isolated dots. The abscissas of these points are natural
numbers: n=1; 2; 3; 4; ... . Ordinates - member values
sequences: a1; a2; a3; a4;…
Example: Write down all five members of a number sequence,
given in a graphical way.
Decision.
Each point in this coordinate plane has
coordinates (n; an). Write down the coordinates of the marked points
ascending abscissa n.
We get: (1; 3), (2; 1), (3; 4), (4; 6), (5; 7).
Therefore, a1= 3; a2=1; a3=4; a4=6; a5=7.

Answer: 3; one; 4; 6; 7.
V. Primary consolidation of the studied material
Example 1. Write a possible formula for the nth element of the sequence (yn):
a) 1, 3, 5, 7, 9, 11, ...;
b) 4, 8, 12, 16, 20, ...;
Decision.
a) This is a sequence odd numbers. Analytically, this sequence can be
set by the formula y = 2n+1.
b) This is a numerical sequence in which the next element is greater than the previous one
by 4. Analytically, this sequence can be given by the formula y = 4n.
Example 2. Write out the first ten elements of a sequence given recurrently: y1=1,
y2=2, yn = yn2+yn1 if n = 3, 4, 5, 6, ... .
Decision.
Each subsequent element of this sequence is equal to the sum of the previous two
elements.
y1=1;
y2=2;
y3=1+2=3;
y4=2+3=5;
y5=3+5=8;
y6=5+8=13;
y7=8+13=21;
y8=13+21=34;
y9=21+34=55;
y10=34+55=89.
VI. Summing up the lesson. Reflection
1. What did you succeed in completing the task?
2. Was the work coordinated?
3. What did not work out, in your opinion?

A numerical sequence is a special case of a numerical function, so a number of properties of functions are also considered for sequences.

1. Definition . Subsequence ( y n} is called increasing if each of its terms (except the first) is greater than the previous one:

y 1 < y 2 < y 3 < … < y n < y n+1 < ….

2. Definition.Sequence ( y n} is called decreasing if each of its terms (except the first) is less than the previous one:

y 1 > y 2 > y 3 > … > y n> y n+1 > … .

3. Increasing and decreasing sequences are united by a common term - monotonic sequences.

For example: y 1 = 1; y n= n 2… is an increasing sequence. y 1 = 1; is a descending sequence. y 1 = 1; – this sequence is not non-increasing non-decreasing.

4. Definition. A sequence is called periodic if there exists a natural number T such that, starting from some n, the equality yn = yn+T holds. The number T is called the period length.

5. A sequence is called bounded from below if all its members are at least some number.

6. A sequence is said to be bounded from above if all its members are at most some number.

7. A sequence is called bounded if it is bounded both above and below, i.e. there is a positive number such that all terms of the given sequence do not exceed this number in absolute value. (But being limited on both sides does not necessarily mean that it is finite.)

8. A sequence can only have one limit.

9. Any non-decreasing sequence bounded above has a limit (lim).

10. Any nonincreasing sequence bounded below has a limit.

The limit of the sequence is a point (number) in the vicinity of which the majority of the members of the sequence are located, they closely approach this limit, but do not reach it.

Geometric and arithmetic progression are special cases of the sequence.

Sequencing methods:

Sequences can be set different ways, among which three are especially important: analytical, descriptive and recurrent.

1. The sequence is given analytically if the formula of its nth member is given:

Example. yn \u003d 2n - 1 - a sequence of odd numbers: 1, 3, 5, 7, 9, ...

2. A descriptive way of setting a numerical sequence is that it explains what elements the sequence is built from.

Example 1. "All members of the sequence are equal to 1." It means, we are talking about the stationary sequence 1, 1, 1, …, 1, ….

Example 2. "The sequence consists of all prime numbers in ascending order." Thus, the sequence 2, 3, 5, 7, 11, … is given. With this method of specifying the sequence in this example it is difficult to answer what, say, the 1000th element of the sequence is equal to.

3. A recurrent way of specifying a sequence is that a rule is indicated that allows one to calculate the nth member of the sequence if its previous members are known. The name recursive method comes from Latin word recurrere - to return. Most often, in such cases, a formula is indicated that allows expressing the nth member of the sequence in terms of the previous ones, and 1–2 initial members of the sequence are specified.

Example 1. y1 = 3; yn = yn–1 + 4 if n = 2, 3, 4,….

Here y1 = 3; y2 = 3 + 4 = 7; y3 = 7 + 4 = 11; ….

It can be seen that the sequence obtained in this example can also be specified analytically: yn = 4n – 1.

Example 2 y 1 = 1; y 2 = 1; y n = y n–2 + y n-1 if n = 3, 4,….

Here: y 1 = 1; y 2 = 1; y 3 = 1 + 1 = 2; y 4 = 1 + 2 = 3; y 5 = 2 + 3 = 5; y 6 = 3 + 5 = 8;

The sequence composed in this example is specially studied in mathematics, since it has a series interesting properties and applications. It is called the Fibonacci sequence - after the Italian mathematician of the 13th century. Defining the Fibonacci sequence recursively is very easy, but analytically it is very difficult. n The th Fibonacci number is expressed in terms of its ordinal number by the following formula.

At first glance, the formula for n th Fibonacci number seems implausible, since the formula that specifies the sequence of natural numbers alone contains square roots, but you can check "manually" the validity of this formula for the first few n.

Fibonacci history:

Fibonacci (Leonardo of Pisa), c. 1175–1250

Italian mathematician. Born in Pisa, became the first great mathematician of Europe in the late Middle Ages. He was led to mathematics by the practical need to establish business contacts. He published his books on arithmetic, algebra and other mathematical disciplines. From Muslim mathematicians, he learned about the system of numbers invented in India and already adopted in the Arab world, and was convinced of its superiority (these numbers were the forerunners of modern Arabic numerals).

Leonardo of Pisa, known as Fibonacci, was the first of the great European mathematicians of the late Middle Ages. Born in Pisa to a wealthy merchant family, he entered mathematics through a purely practical need to establish business contacts. In his youth, Leonardo traveled a lot, accompanying his father on business trips. For example, we know about his long stay in Byzantium and Sicily. During such trips, he interacted a lot with local scientists.

The number sequence that bears his name today grew out of the problem with rabbits that Fibonacci outlined in his Liber abacci, written in 1202:

A man put a pair of rabbits in a pen, surrounded on all sides by a wall. How many pairs of rabbits can this pair give birth to in a year, if it is known that every month, starting from the second, each pair of rabbits produces one pair?

You can make sure that the number of couples in each of the next twelve months of the months will be respectively 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

In other words, the number of pairs of rabbits creates a series, each term in which is the sum of the previous two. It is known as the Fibonacci series and the numbers themselves are the Fibonacci numbers. It turns out that this sequence has many mathematically interesting properties. Here's an example: you can divide a line into two segments so that the ratio between the larger and smaller segment is proportional to the ratio between the entire line and the larger segment. This proportionality factor, approximately equal to 1.618, is known as golden ratio. In the Renaissance, it was believed that this proportion, observed in architectural structures, is most pleasing to the eye. If you take consecutive pairs from the Fibonacci series and divide more from each pair to a smaller one, your result will gradually approach the golden ratio.

Since Fibonacci discovered his sequence, even natural phenomena have been found in which this sequence seems to play an important role. One of them is phyllotaxis (leaf arrangement) - the rule according to which, for example, seeds are located in a sunflower inflorescence. Sunflower seeds are arranged in two spirals. The numbers indicating the number of seeds in each of the spirals are members of an amazing mathematical sequence. The seeds are arranged in two rows of spirals, one of which goes clockwise, the other against. And what is the number of seeds in each case? 34 and 55.

Task #1:

Write the first five terms of the sequence.

1. a n \u003d 2 n + 1/2 n

and n \u003d 2 n + 1/2 n

Task number 2:

Write the formula for the common term of a sequence of natural numbers that are multiples of 3.

Answer: 0,3,6,9,12,15,.... 3n, and n = 3n

Task number 3:

Write the formula for the common term of a sequence of natural numbers that, when divided by 4, have a remainder of 1.

Answer: 5,9,13,17,21....... 4 n +1 and n = 4n+1

No. 19. Function.

Function (display, operator, transformation) is a mathematical concept that reflects the relationship between elements of sets. We can say that a function is a "law" according to which each element of one set (called the domain of definition) is assigned some element of another set (called the domain of values).

A function is a dependency of one variable from another. In other words, the relationship between quantities.

The mathematical concept of a function expresses an intuitive idea of ​​how one quantity completely determines the value of another quantity. So the value of the variable x uniquely determines the value of the expression, and the value of the month uniquely determines the value of the month following it, and any person can be compared to another person - his father. Similarly, some pre-conceived algorithm, given varying input data, produces certain output data.

Often the term "function" refers to a numerical function; that is, a function that puts some numbers in correspondence with others. These functions are conveniently represented in the figures in the form of graphs.

Another definition can be given. A function is a specific action over a variable.

This means that we take the value , do some action with it (for example, we square it or calculate its logarithm) - and we get the value .

Let's give another definition of a function - the one that is most often found in textbooks.

A function is a correspondence between two sets, with each element of the first set corresponding to one and only one element of the second set.

For example, a function for each real number matches a number twice as large as .

The set of elements of some F. substituted for x is called its domain of definition, and the set of elements y of some F. is called its range of values.

Term history:

The term "function" (in a somewhat narrower sense) was first used by Leibniz (1692). In turn, Johann Bernoulli, in a letter to the same Leibniz, used this term in a sense closer to the modern one. Initially, the concept of a function was indistinguishable from the concept of an analytic representation. Subsequently, the definition of the function given by Euler (1751) appeared, then - by Lacroix (1806) - almost in modern form. Finally, the general definition of a function (in modern form, but for numerical functions) was given by Lobachevsky (1834) and Dirichlet (1837). To late XIX century, the concept of a function has outgrown the framework of numerical systems. Vector functions were the first to do this, Frege soon introduced logical functions (1879), and after the advent of set theory, Dedekind (1887) and Peano (1911) formulated the modern universal definition.

No. 20. Ways to set a function.

There are 4 ways to define a function:

1. tabular Quite common, is to set a table of individual

argument values ​​and their corresponding function values. This method of defining a function is used when the domain of the function is a discrete finite set.

It is convenient when f is a finite set, but when f is infinite, only selected pairs (x, y) are indicated.

With the tabular method of defining a function, it is possible to approximately calculate the values ​​of the function that are not contained in the table, corresponding to the intermediate values ​​of the argument. To do this, use the method of interpolation.

Advantages: accuracy, speed, easy to find in the table of values desired value functions. The advantages of the tabular way of specifying a function are that it makes it possible to determine certain specific values ​​​​at once, without additional measurements or calculations.

disadvantages: incompleteness, lack of clarity. In some cases, the table does not define the function completely, but only for some values ​​of the argument and does not provide a visual representation of the nature of the change in the function depending on the change in the argument.

2. analytical(formulas). Most often, a law establishing a connection between

argument and function, is specified by means of formulas. This way of defining a function is called analytical. It is the most important for MA (math. analysis), since the methods of MA (differential, integral calculus) suggest this way of setting. The same function can be given by different formulas: y=∣sin( x)∣y=√1−cos2( x) Sometimes in various parts of its domains, the defined function can be given by various formulas f(x)={f 1(x),x∈D 1 fn(x),x∈Dn ∪nk=1Dk=D(f) . Often, with this method of defining a function, the scope of definition is not indicated, then the domain of definition is understood as natural area definitions, i.e. the set of all x values ​​for which the function takes a real value.

This method makes it possible for each numerical value of the argument x to find the corresponding numerical value of the function y exactly or with some accuracy.

A special case of the analytical way of defining a function is defining a function by an equation of the form F(x,y)=0 (1) If this equation has the property that ∀ x∈D is matched only y, such that F(x,y)=0, then we say that equation (1) on D implicitly defines a function. Another particular case of defining a function is parametric, with each pair ( x,y)∈f set using a pair of functions x=ϕ( t),y=ψ( t) where t∈M.

The definition of a numerical sequence is given. Examples of infinitely increasing, convergent, and divergent sequences are considered. A sequence containing all rational numbers is considered.

Definition .
Numerical sequence ( x n ) called the law (rule), according to which, for each natural number n = 1, 2, 3, . . . some number x n is assigned.
The element x n is called nth member or an element of a sequence.

The sequence is denoted as the nth member enclosed in curly brackets: . Also possible the following notation: . They explicitly state that the index n belongs to the set of natural numbers and that the sequence itself has an infinite number of members. Here are some examples of sequences:
, , .

In other words, a numerical sequence is a function whose domain is the set of natural numbers. The number of elements in the sequence is infinite. Among the elements, there may also be members that have same values. Also, the sequence can be considered as a numbered set of numbers, consisting of an infinite number of members.

We will be mainly interested in the question - how sequences behave when n tends to infinity: . This material is presented in the Sequence limit - basic theorems and properties section. And here we will look at some examples of sequences.

Sequence examples

Examples of infinitely increasing sequences

Let's consider a sequence. The general term of this sequence is . Let's write out the first few terms:
.
It can be seen that as the number n grows, the elements increase indefinitely towards positive values. We can say that this sequence tends to : at .

Now consider a sequence with a common term . Here are some of its first members:
.
As the number n grows, the elements of this sequence increase in absolute value indefinitely, but do not have a constant sign. That is, this sequence tends to : at .

Examples of sequences converging to a finite number

Let's consider a sequence. Its common member The first terms are as follows:
.
It can be seen that as the number n grows, the elements of this sequence approach their limit value a = 0 : at . So each subsequent term is closer to zero than the previous one. In a sense, we can assume that there is an approximate value for the number a = 0 with an error. It is clear that as n grows, this error tends to zero, that is, by choosing n, the error can be made arbitrarily small. Moreover, for any given error ε > 0 it is possible to specify such a number N , that for all elements with numbers greater than N : , the deviation of the number from the limit value a will not exceed the error ε : .

Next, consider the sequence. Its common member Here are some of its first members:
.
In this sequence, even-numbered terms are zero. Members with odd n are . Therefore, as n grows, their values ​​approach the limiting value a = 0 . This also follows from the fact that
.
As in the previous example, we can specify an arbitrarily small error ε > 0 , for which it is possible to find such a number N that elements with numbers greater than N will deviate from the limit value a = 0 by a value not exceeding the specified error. Therefore, this sequence converges to the value a = 0 : at .

Examples of divergent sequences

Consider a sequence with the following common term:

Here are its first members:


.
It can be seen that the terms with even numbers:
,
converge to the value a 1 = 0 . Members with odd numbers:
,
converge to the value a 2 = 2 . The sequence itself, as n grows, does not converge to any value.

Sequence with terms distributed in the interval (0;1)

Now consider a more interesting sequence. Take a segment on the number line. Let's split it in half. We get two segments. Let be
.
Each of the segments is again divided in half. We get four segments. Let be
.
Divide each segment in half again. Let's take


.
Etc.

As a result, we obtain a sequence whose elements are distributed in an open interval (0; 1) . Whatever point we take from the closed interval , we can always find members of the sequence that are arbitrarily close to this point, or coincide with it.

Then from the original sequence one can single out a subsequence that will converge to an arbitrary point from the interval . That is, as the number n grows, the members of the subsequence will come closer and closer to the preselected point.

For example, for point a = 0 you can choose the following subsequence:
.
= 0 .

For point a = 1 choose the following subsequence:
.
The members of this subsequence converge to the value a = 1 .

Since there are subsequences that converge to different meanings, then the original sequence itself does not converge to any number.

Sequence containing all rational numbers

Now let's construct a sequence that contains all rational numbers. Moreover, each rational number will be included in such a sequence an infinite number of times.

The rational number r can be represented as follows:
,
where is an integer; - natural.
We need to assign to each natural number n a pair of numbers p and q so that any pair of p and q is included in our sequence.

To do this, draw axes p and q on the plane. We draw grid lines through integer values ​​p and q . Then each node of this grid with will correspond to rational number. The whole set of rational numbers will be represented by a set of nodes. We need to find a way to number all the nodes so that we don't miss a single node. This is easy to do if we number the nodes according to the squares whose centers are located at the point (0; 0) (see picture). In this case, the lower parts of the squares with q < 1 we don't need. Therefore, they are not shown in the figure.

So, for the upper side of the first square we have:
.
Further we number upper part next square:

.
We number the upper part of the next square:

.
Etc.

In this way we get a sequence containing all rational numbers. It can be seen that any rational number appears in this sequence an infinite number of times. Indeed, along with the node , this sequence will also include nodes , where is a natural number. But all these nodes correspond to the same rational number.

Then from the sequence we have constructed, we can select a subsequence (having an infinite number of elements), all elements of which are equal to a predetermined rational number. Since the sequence we have constructed has subsequences converging to different numbers, then the sequence does not converge to any number.

Conclusion

Here we have given a precise definition of the numerical sequence. We also touched upon the issue of its convergence, based on intuitive ideas. The exact definition of convergence is discussed on the page Determining the Limit of a Sequence. Related properties and theorems are outlined on the page

Lesson #32 Date ____________

Algebra

Class: 9 "B"

Topic: "Numerical sequence and ways to set it."

The purpose of the lesson: students should know what a number sequence is; ways to set a numerical sequence; be able to distinguish between different ways of specifying numerical sequences.

Didactic materials: handouts, reference notes.

Technical means learning: presentation on the topic "Numeric sequences".

During the classes.

1. Organizational moment.

2. Setting the goals of the lesson.

Today in the lesson you guys will learn:

What is a sequence?

What kinds of sequences are there?

How is the number sequence specified?

Learn how to write a sequence using a formula and its many elements.

Learn to find the members of a sequence.

3. Work on the studied material.

3.1. Preparatory stage.

Guys, let's test your logic skills. I name a few words, and you should continue:

-Monday Tuesday,…..

- January February March…;

- Glebova L, Ganovichev E, Dryakhlov V, Ibraeva G, ... .. (class list);

–10,11,12,…99;

From the answers of the guys, it is concluded that the above tasks are sequences, that is, some sort of ordered series of numbers or concepts, when each number or concept is strictly in its place, and if the members are swapped, the sequence will be violated (Tuesday, Thursday, Monday is just a list of the days of the week). So, the topic of the lesson is a numerical sequence.

3.1. Explanation of new material. (Demo material)

Analyzing students' answers, define the number sequence and show how to set number sequences.

(Working with the textbook pp. 66 - 67)

Definition 1. The function y = f(x), xN is called a function of a natural argument or a numerical sequence and denoted: y = f(n) or y 1 , y 2 , y 3 , ..., y n , ... or (y n).

In this case, the independent variable is a natural number.

Most often, sequences will be denoted as follows: ( a n), (b n), (with n) etc.

Definition 2. Sequence Members.

The elements that form a sequence are called members of the sequence.

New concepts: the previous and subsequent member of the sequence,

a 1 …a P. (1st and nth member of the sequence)

Methods for setting a numerical sequence.

analytical way.

Any nth element sequences can be determined using a formula. (demo)

Parse examples

Example 1 The sequence of even numbers: y = 2n.

Example 2 The sequence of the square of natural numbers: y = n 2 ;

1, 4, 9, 16, 25, ..., n 2 , ... .

Example 3 Stationary sequence: y = C;

C, C, C, ..., C, ... .

Special case: y = 5; 5, 5, 5, ..., 5, ... .

Example 4. Sequence y = 2 n ;

2, 2 2 , 2 3 , 2 4 , ..., 2 n , ... .

verbal way.

The rules for setting the sequence are described in words, without specifying formulas or when there are no patterns between the elements of the sequence.

Example 1. Number Approximationsπ.

Example 2 Prime number sequence: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, .... .

Example 3 A sequence of numbers divisible by 5.

Example 2 Random set of numbers: 1, 4, 12, 25, 26, 33, 39, ... .

Example 3 The sequence of even numbers 2, 4, 6, 8, 10, 12, 14, 16, ... .

recursive way.

The recurrent method consists in specifying a rule that allows you to calculate the nth member of the sequence if its first few members are specified (at least one first member) and a formula that allows you to calculate its next member from the previous members. Term recurrent derived from the Latin word recurrere , which means come back . When calculating the members of the sequence according to this rule, we kind of go back all the time, calculating the next member based on the previous one. A feature of this method is that to determine, for example, the 100th member of the sequence, you must first determine all the previous 99 members.

Example 1 . a 1 \u003d a, a n + 1 \u003d a n +0.7. Let a 1 =5, then the sequence will look like: 5; 5.7; 6.4; 7.1; 7.8; 8.5; ... .

Example 2 b 1 \u003d b, b n +1 \u003d ½ b n. Let b 1 =23, then the sequence will look like: 23; 11.5; 5.75; 2.875; ... .

Example 3 Fibonacci sequence. This sequence is easily defined recursively: y 1 =1, y 2 =1,y n -2 +y n -1 if n=3, 4, 5, 6, ... . It will look like:

1, 1,2, 3, 5, 8, 13, 21, 34, 55, ... . (P th term of this sequence is equal to the sum of the two previous terms)

It is difficult to define the Fibonacci sequence analytically, but it is possible. The formula by which any element of this sequence is determined looks like this:

Additional Information:

The Italian merchant Leonardo of Pisa (1180-1240), better known by the nickname Fibonacci, was an important medieval mathematician. With the help of this sequence, Fibonacci determined the number φ (phi); φ=1.618033989.

Graphical way

The members of a sequence can be represented as points on the coordinate plane. To do this, the number is plotted along the horizontal axis, and the value of the corresponding member of the sequence is plotted along the vertical axis.

To consolidate the methods of assignment, I ask you to give several examples of sequences that are specified either verbally, or analytically, or in a recurrent way.

Types of number sequences

(On the sequences listed below, types of sequences are worked out).

Working with the textbook p.69-70

1) Increasing - if each term is less than the next one, i.e. a n a n +1.

2) Decreasing - if each term is greater than the next one, i.e. a n a n +1 .

3) Endless.

4) Ultimate.

5) Alternating.

6) Constant (stationary).

An increasing or decreasing sequence is called monotonic.

3; 6; 9; 12; 15; 18;…

1. –1; 2; –3; 4; –5; …

   1, 4, 9, 16 ,…

   –1; 2; –3; 4; –5; 6; …

   3; 3; 3; 3; …; 3; … .

Work with the textbook: do it orally No. 150, 159 pp. 71, 72

3.2. Consolidation of new material. Problem solving.

To consolidate knowledge, examples are selected depending on the level of preparation of students.

Example 1 Write a possible formula for the nth element of the sequence (y n):

a) 1, 3, 5, 7, 9, 11, ...;

b) 4, 8, 12, 16, 20, ...;

Decision.

a) It is a sequence of odd numbers. Analytically, this sequence can be given by the formula y = 2n+1.

b) This is a numerical sequence in which the next element is greater than the previous one by 4. Analytically, this sequence can be specified by the formula y = 4n.

Example 2. Write out the first ten elements of the sequence given recurrently: y 1 =1, y 2 =2, y n = y n -2 +y n -1 if n = 3, 4, 5, 6, ... .

Decision.

Each subsequent element of this sequence is equal to the sum of the two previous elements.

Example 3 The sequence (y n) is given recurrently: y 1 =1, y 2 =2,y n =5y n -1 - 6y n -2 . Specify this sequence analytically.

Decision.

Find the first few elements of the sequence.

y 3 =5y 2 -6y 1 =10-6=4;

y 4 \u003d 5y 3 -6y 2 \u003d 20-12 \u003d 8;

y 5 \u003d 5y 4 -6y 3 \u003d 40-24 \u003d 16;

y 6 \u003d 5y 5 -6y 4 \u003d 80-48 \u003d 32;

y 7 \u003d 5y 6 -6y 5 \u003d 160-96 \u003d 64.

We get the sequence: 1; 2; 4; eight; sixteen; 32; 64; ... which can be represented as

2 0 ; 2 1 ; 2 2 ; 2 3 ; 2 4 ; 2 5 ; 2 6 ... .

n = 1; 2; 3; 4; 5; 6; 7... .

Analyzing the sequence, we obtain the following regularity: y = 2 n -1 .

Example 4 Given a sequence y n =24n+36-5n 2 .

a) How many positive terms does it have?

b) Find the largest element of the sequence.

c) Is there a smallest element in this sequence?

This numerical sequence is a function of the form y = -5x 2 +24x+36, where x

a) Find the values ​​of the function for which -5x 2 +24x+360. Let's solve the equation -5x 2 +24x+36=0.

D \u003d b 2 -4ac \u003d 1296, X 1 \u003d 6, X 2 \u003d -1.2.

The equation of the axis of symmetry of the parabola y \u003d -5x 2 +24x + 36 can be found by the formula x \u003d, we get: x \u003d 2.4.

The inequality -5x 2 +24x+360 holds for -1.2 This interval contains five natural numbers (1, 2, 3, 4, 5). So in the given sequence five positive elements sequences.

b) The largest element of the sequence is determined by the selection method and it is equal to y 2 =64.

c) There is no smallest element.

3.4. Tasks for independent work