Consider a series of natural numbers: 1, 2, 3, , n – 1, n,  .

If we replace every natural number n in this series some number a n, following some law, we get a new series of numbers:

a 1 , a 2 , a 3 , , a n –1 , a n , ,

abbreviated and called numerical sequence. Value a n is called the common member of the numerical sequence. Usually the numerical sequence is given by some formula a n = f(n) that allows you to find any member of the sequence by its number n; this formula is called the general term formula. Note that it is not always possible to specify a numerical sequence by a general term formula; sometimes a sequence is specified by describing its members.

By definition, a sequence always contains an infinite number of elements: any two different elements of it differ at least in their numbers, of which there are infinitely many.

The numeric sequence is a special case of a function. A sequence is a function defined on the set of natural numbers and taking values ​​in the set of real numbers, i.e., a function of the form f : N  R.

Subsequence
called increasing(waning), if for any n  N
Such sequences are called strictly monotonous.

Sometimes it is convenient to use as numbers not all natural numbers, but only some of them (for example, natural numbers starting from some natural number n 0). For numbering, it is also possible to use not only natural numbers, but also other numbers, for example, n= 0, 1, 2,  (here, zero is added to the set of natural numbers as another number). In such cases, specifying the sequence, indicate what values ​​the numbers take. n.

If in some sequence for any n  N
then the sequence is called non-decreasing(non-increasing). Such sequences are called monotonous.

Example 1 . The numerical sequence 1, 2, 3, 4, 5, ... is a series of natural numbers and has a common term a n = n.

Example 2 . The number sequence 2, 4, 6, 8, 10, ... is a series of even numbers and has a common term a n = 2n.

Example 3 . 1.4, 1.41, 1.414, 1.4142, … is a numerical sequence of approximate values ​​with increasing accuracy.

In the last example, it is impossible to give a formula for the common term of the sequence.

Example 4 . Write the first 5 terms of a numerical sequence by its common term
. To calculate a 1 is needed in the formula for the common term a n instead of n substitute 1 to calculate a 2 − 2, etc. Then we have:

Test 6 . The common member of the sequence 1, 2, 6, 24, 120,  is:

1)

2)

3)

4)

Test 7 .
is:

1)

2)

3)

4)

Test 8 . Common Member of the Sequence
is:

1)

2)

3)

4)

Number Sequence Limit

Consider a numerical sequence whose common term approaches a certain number BUT with increasing serial number n. In this case, the number sequence is said to have a limit. This concept has a more rigorous definition.

Number BUT is called the limit of the number sequence
:

(1)

if for any  > 0 there is such a number n 0 = n 0 (), depending on , which
at n > n 0 .

This definition means that BUT there is a limit of a number sequence if its common term indefinitely approaches BUT with increasing n. Geometrically, this means that for any  > 0 one can find such a number n 0 , which, starting from n > n 0 , all members of the sequence are located inside the interval ( BUT – , BUT+ ). A sequence that has a limit is called converging; otherwise - divergent.

A number sequence can have only one limit (finite or infinite) of a certain sign.

Example 5 . Harmonic sequence has the number 0 as a limit. Indeed, for any interval (–; +) as a number N 0 can be any integer greater than . Then for all n > n 0 > we have

Example 6 . The sequence 2, 5, 2, 5,  is divergent. Indeed, no interval of length less than, for example, one, can contain all members of the sequence, starting from some number.

The sequence is called limited if there is such a number M, what
for all n. Every convergent sequence is bounded. Every monotone and bounded sequence has a limit. Every convergent sequence has a unique limit.

Example 7 . Subsequence
is increasing and limited. She has a limit
=e.

Number e called Euler number and is approximately equal to 2.718 28.

Test 9 . The sequence 1, 4, 9, 16,  is:

1) converging;

2) divergent;

3) limited;

Test 10 . Subsequence
is:

1) converging;

2) divergent;

3) limited;

4) arithmetic progression;

5) geometric progression.

Test 11 . Subsequence is not:

1) converging;

2) divergent;

3) limited;

4) harmonic.

Test 12 . Limit of the sequence given by the common term
equal.

The numerical sequence and its limit are one of the most important problems of mathematics throughout the history of the existence of this science. Constantly updated knowledge, formulated new theorems and proofs - all this allows us to consider this concept from new positions and under different

A numerical sequence, in accordance with one of the most common definitions, is a mathematical function, the basis of which is the set of natural numbers arranged according to one pattern or another.

There are several options for creating number sequences.

First, this function can be specified in the so-called "explicit" way, when there is a certain formula by which each of its members can be determined by simply substituting the ordinal number into the given sequence.

The second method is called "recursive". Its essence lies in the fact that the first few members of the numerical sequence are given, as well as a special recursive formula, with the help of which, knowing the previous member, you can find the next one.

Finally, the most general way of specifying sequences is the so-called when, without much difficulty, one can not only identify one or another term under a certain serial number, but also, knowing several consecutive terms, come to the general formula of this function.

The numerical sequence can be decreasing or increasing. In the first case, each subsequent term is less than the previous one, and in the second, on the contrary, it is greater.

Considering this topic, it is impossible not to touch on the issue of the limits of sequences. The limit of a sequence is such a number when for any, including for an infinitely small value, there is an ordinal number, after which the deviation of successive members of the sequence from a given point in numerical form becomes less than the value specified during the formation of this function.

The concept of the limit of a numerical sequence is actively used when carrying out certain integral and differential calculations.

Mathematical sequences have a whole set of rather interesting properties.

Firstly, any numerical sequence is an example of a mathematical function, therefore, those properties that are characteristic of functions can be safely applied to sequences. The most striking example of such properties is the provision on increasing and decreasing arithmetic series, which are united by one common concept - monotonic sequences.

Secondly, there is a fairly large group of sequences that cannot be classified as either increasing or decreasing - these are periodic sequences. In mathematics, they are considered to be those functions in which there is a so-called period length, that is, from a certain moment (n), the following equality begins to operate y n \u003d y n + T, where T will be the very length of the period.

Cradle. Diapers. Cry.
Word. Step. Cold. Doctor.
running around. Toys. Brother.
Yard. Swing. Kindergarten.
School. Deuce. Troika. Five.
Ball. Step. Gypsum. Bed.
Fight. Blood. Broken nose.
Yard. Friends. Party. Force.
Institute. Spring. bushes.
Summer. Session. Tails.
Beer. Vodka. Iced gin.
Coffee. Session. Diploma.
Romanticism. Love. Star.
Arms. Lips. Night without sleep.
Wedding. Mother-in-law. Father-in-law. Trap.
Argument. Club. Friends. Cup.
House. Job. House. A family.
Sun. Summer. Snow. Winter.
Son. Diapers. Cradle.
Stress. Mistress. Bed.
Business. Money. Plan. Avral.
Television. TV series.
Country house. Cherries. Zucchini.
Gray hair. Migraine. Glasses.
Grandson. Diapers. Cradle.
Stress. Pressure. Bed.
Heart. Kidneys. Bones. Doctor.
Speeches. Coffin. Farewell. Cry.

life sequence

SEQUENCE - (sequence), numbers or elements arranged in an organized manner. Sequences can be finite (having a limited number of elements) or infinite, like a complete sequence of natural numbers 1, 2, 3, 4 ….… …

Scientific and technical encyclopedic dictionary

Definition:Numerical sequence is called numerical, given on the set N of natural numbers. For numerical sequences, usually instead of f(n) write a n and denote the sequence like this: a n ). Numbers a 1 , a 2 , …, a n,… called sequence elements.

Usually the numerical sequence is determined by setting n-th element or a recursive formula, according to which each next element is determined through the previous one. A descriptive way of specifying a numerical sequence is also possible. For example:

• All members of the sequence are "1". This means that we are talking about a stationary sequence 1, 1, 1, …, 1, ….

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

With the recurrent method, a formula is indicated that allows you to express n th member of the sequence through the previous ones, and 1–2 initial members of the sequence are specified.

• y 1 = 3; y n =y n-1 + 4 , if n = 2, 3, 4,…

Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; ….

• 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;

Sequence expressed by recursive formula y n =y n-1 + 4 can also be given analytically: y n= y 1 +4*(n-1)

Check: y2=3+4*(2-1)=7, y3=3+4*(3-1)=11

Here we do not need to know the previous member of the numerical sequence to calculate the nth element, it is enough just to set its number and the value of the first element.

As we can see, this way of specifying a numerical sequence is very similar to the analytical way of specifying functions. In fact, a numerical sequence is a special kind of a numerical function, so a number of properties of functions can be considered for sequences as well.

Number sequences are a very interesting and informative topic. This topic is found in tasks of increased complexity, which are offered to students by the authors of didactic materials, in the tasks of mathematical olympiads, entrance exams to higher educational institutions and on. And if you want to learn more about the different types of number sequences, click here. Well, if everything is clear and simple for you, but try to answer.

Hovhannisyan Eva

Numeric sequences. Abstract.

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Municipal budgetary educational institution
"Secondary school No. 31"
the city of Barnaul

Number Sequences

abstract

Work completed:
Oganesyan Eva,
8th grade student MBOU "Secondary School No. 31"
Supervisor:
Poleva Irina Alexandrovna,
mathematics teacher MBOU "Secondary School No. 31"

Barnaul - 2014

Introduction…………………………………………………………………………2

Numerical sequences.……………………………………………...3

Ways to set numerical sequences………………………...4

Development of the doctrine of progressions………………………………………………..5

Properties of numerical sequences……………………………………7

Arithmetic progression……………………………...................................................9

Geometric progression……………………………………………….10

Conclusion ………………………………………………………………… 11

References……………………………………………………………11

Introduction

Purpose of this abstract– study of the basic concepts related to numerical sequences, their application in practice.
Tasks:

1. To study the historical aspects of the development of the doctrine of progressions;
2. Consider ways of setting and properties of numerical sequences;
3. Learn about arithmetic and geometric progressions.

Currently, numerical sequences are considered as special cases of a function. The numerical sequence is a function of the natural argument. The concept of a numerical sequence arose and developed long before the creation of the theory of function. Here are examples of infinite number sequences known in antiquity:

1, 2, 3, 4, 5, … - sequence of natural numbers.

2, 4, 6, 8, 10,… - a sequence of even numbers.

1, 3, 5, 7, 9,… - a sequence of odd numbers.

1, 4, 9, 16, 25,… - sequence of squares of natural numbers.

2, 3, 5, 7, 11… - a sequence of prime numbers.

1, ½, 1/3, ¼, 1/5,… - sequence of reciprocals of natural numbers.

The number of members of each of these series is infinite; the first five sequences are monotonically increasing, the last one is monotonically decreasing. All of the listed sequences, except for the 5th one, are given due to the fact that for each of them the common term is known, i.e., the rule for obtaining a term with any number. For a sequence of prime numbers, a common term is unknown, but as early as the 3rd century. BC e. the Alexandrian scientist Eratosthenes indicated a method (albeit very cumbersome) for obtaining its n-th member. This method was called the "sieve of Eratosthenes".

Progressions - particular types of numerical sequences - are found in the monuments of the II millennium BC. e.

Number Sequences

There are various definitions of the number sequence.

Numeric sequence – it is a sequence of elements of a number space (Wikipedia).

Numeric sequence – this is a numbered number set.

A function of the form y = f (x), xis called a function of natural argument ornumerical sequenceand denote y = f(n) or

, , , …, The notation ().

We will write out positive even numbers in ascending order. The first such number is 2, the second is 4, the third is 6, the fourth is 8, and so on, so we get the sequence: 2; four; 6; eight; ten ….

Obviously, the fifth place in this sequence will be the number 10, the tenth - 20, the hundredth - 200. In general, for any natural number n, you can specify the corresponding positive even number; it is equal to 2n.

Let's look at another sequence. We will write out in descending order proper fractions with a numerator equal to 1:

; ; ; ; ; … .

For any natural number n, we can specify the corresponding fraction; it is equal to. So, in sixth place should be a fraction, on the thirtieth - , on the thousandth - a fraction .

The numbers that form a sequence are called the first, second, third, fourth, etc., respectively. members of the sequence. The members of a sequence are usually denoted by letters with subscripts indicating the ordinal number of the member. For example:, , etc. in general, the term of the sequence with number n, or, as they say, the nth member of the sequence, is denoted. The sequence itself is denoted by (). A sequence can contain both an infinite number of members and a finite one. In this case, it is called final. For example: a sequence of two-digit numbers.10; eleven; 12; 13; …; 98; 99

Methods for specifying numerical sequences

Sequences can be specified in several ways.

Usually the sequence is more appropriate to setformula of its common nth term, which allows you to find any member of the sequence, knowing its number. In this case, the sequence is said to be given analytically. For example: a sequence of positive even terms=2n.

A task: find the formula for the common term of the sequence (:

6; 20; 56; 144; 352;…

Solution. We write each term of the sequence in the following form:

n=1: 6 = 2 3 = 3 =

n=2: 20=4 5=5=

n=3: 56 = 8 7 = 7 =

As you can see, the terms of the sequence are the product of a power of two multiplied by consecutive odd numbers, and two is raised to a power that is equal to the number of the element in question. Thus, we conclude that

Answer: common term formula:

Another way to specify a sequence is to specify a sequence usingrecurrent relation. A formula that expresses any member of a sequence, starting with some through the previous ones (one or more), is called recurrent (from the Latin word recurro - to return).

In this case, one or several first elements of the sequence are specified, and the rest are determined according to some rule.

An example of a recursively given sequence is the sequence of Fibonacci numbers - 1, 1, 2, 3, 5, 8, 13, ... , in which each subsequent number, starting from the third, is the sum of the two previous ones: 2 = 1 + 1; 3 = 2 + 1 and so on. This sequence can be given recursively:

N N, = 1.

A task: subsequencegiven by the recurrence relation+ , n N, = 4. Write down the first few terms of this sequence.

Solution. Let's find the third term of the given sequence:

+ =

Etc.

When sequences are specified recurrently, calculations are very cumbersome, since in order to find elements with large numbers, it is necessary to find all previous members of the specified sequence, for example, to findwe need to find all the previous 499 members.

Descriptive wayassignment of a numerical sequence consists in explaining what elements the sequence is built from.

Example 1 . "All members of the sequence are 1." This means that we are talking about a 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 way of specifying the sequence in this example, it is difficult to answer what, say, the 1000th element of the sequence is equal to.

Also, a numerical sequence can be given by a simplelisting its members.

Development of the doctrine of progressions

The word progression is of Latin origin (progressio), literally means “moving forward” (like the word “progress”) and is first encountered by the Roman author Boethius (5th-6th centuries). continue it indefinitely in one direction, for example, a sequence of natural numbers, their squares and cubes. At the end of the Middle Ages and at the beginning of modern times, this term ceases to be commonly used. In the 17th century, for example, J. Gregory used the term "series" instead of progression, and another prominent English mathematician, J. Wallis, used the term "infinite progressions" for infinite series.

At present, we consider progressions as special cases of numerical sequences.

Theoretical information related to progressions is first found in the documents of ancient Greece that have come down to us.

In the Psammite, Archimedes for the first time compares arithmetic and geometric progressions:

1,2,3,4,5,………………..

10, , ………….

Progressions were considered as a continuation of proportions, which is why the epithets arithmetic and geometric were transferred from proportions to progressions.

This view of progressions was preserved by many mathematicians of the 17th and even 18th centuries. This is how one should explain the fact that the symbol found in Barrow, and then in other English scientists of that time to denote a continuous geometric proportion, began to denote a geometric progression in English and French textbooks of the 18th century. By analogy, they began to designate an arithmetic progression.

One of the proofs of Archimedes, set forth in his work “The Quadrature of the Parabola”, essentially boils down to the summation of an infinitely decreasing geometric progression.

To solve some problems from geometry and mechanics, Archimedes derived the formula for the sum of the squares of natural numbers, although it was used before him.

1/6n(n+1)(2n+1)

Some formulas related to progressions were known to Chinese and Indian scientists. So, Aryabhatta (V century) knew the formulas for the common term, the sum of an arithmetic progression, etc., Magavira (IX century) used the formula: + + + ... + = 1/6n(n+1)(2n+1) and other more complex series. However, the rule for finding the sum of terms of an arbitrary arithmetic progression is first found in the Book of the Abacus (1202) by Leonardo of Pisa. In The Science of Numbers (1484), N. Shuke, like Archimedes, compares the arithmetic progression with the geometric one and gives a general rule for summing any infinitely small decreasing geometric progression. The formula for summing an infinitely decreasing progression was known to P. Fermat and other mathematicians of the 17th century.

Problems for arithmetic (and geometric) progressions are also found in the ancient Chinese tract "Mathematics in Nine Books", which, however, does not contain instructions for the use of any summation formula.

The first progression problems that have come down to us are connected with the demands of economic life and social practice, such as the distribution of products, the division of inheritance, and so on.

From one cuneiform tablet, we can conclude that, observing the moon from new moon to full moon, the Babylonians came to the following conclusion: in the first five days after the new moon, the increase in illumination of the lunar disk occurs according to the law of geometric progression with a denominator of 2. In another later tablet, we are talking about summation geometric progression:

1+2+ +…+ . solution and answer S=512+(512-1), the data in the plate suggest that the author used the formula.

Sn= +( -1), but no one knows how he reached it.

The summation of geometric progressions and the compilation of corresponding problems that do not always meet practical needs were practiced by many lovers of mathematics throughout the ancient and middle ages.

Number Sequence Properties

A numerical sequence is a special case of a numerical function, and therefore some properties of functions (boundedness, monotonicity) are also considered for sequences.

Limited sequences

Subsequence () is called bounded from above, that for any number n, M.

Subsequence () is called bounded from below, if there is such a number m, that for any number n, m.

Subsequence () is called bounded , if it is bounded from above and bounded from below, that is, there exists such a number M0 , which for any number n , M.

Subsequence () is called unbounded , if there exists such a number M0 that there exists a number n such that, M.

A task: explore the sequence = to limitation.

Solution. The given sequence is bounded, since for any natural number n the following inequalities hold:

0 1,

That is, the sequence is bounded from below by zero, and at the same time it is bounded from above by one, and therefore, it is also bounded.

Answer: the sequence is limited - from below by zero, and from above by one.

Increasing and descending sequences

Subsequence () is called increasing , if each term is greater than the previous one:

For example, 1, 3, 5, 7.....2n -1,... is an increasing sequence.

Subsequence () is called decreasing , if each term is less than the previous one:

For example, 1; is a descending sequence.

Increasing and decreasing sequences are combined by a common term -monotonic sequences. Let's take a few more examples.

1; - this sequence is neither increasing nor decreasing (nonmonotonic sequence).

2n. We are talking about the sequence 2, 4, 8, 16, 32, ... - an increasing sequence.

In general, if a > 1, then the sequence= increases;

if 0 = decreasing.

Arithmetic progression

A numerical sequence, each member of which, starting from the second, is equal to the sum of the previous member and the same number d, is calledarithmetic progression, and the number d is the difference of an arithmetic progression.

Thus, an arithmetic progression is a numerical sequence

X, == + d, (n = 2, 3, 4, …; a and d are given numbers).

Example 1. 1, 3, 5, 7, 9, 11, ... is an increasing arithmetic progression, in which= 1, d = 2.

Example 2. 20, 17, 14, 11, 8, 5, 2, -1, -4, ... - a decreasing arithmetic progression, in which= 20, d = –3.

Example 3. Consider a sequence of natural numbers that, when divided by four, have a remainder of 1: 1; 5; 9; 13; 17; 21…

Each of its members, starting from the second, is obtained by adding the number 4 to the previous member. This sequence is an example of an arithmetic progression.

It is easy to find an explicit (formula) expressionthrough n. The value of the next element increases by d compared to the previous one, thus, the value of the n element will increase by (n - 1)d compared to the first member of the arithmetic progression, i.e.

= + d (n – 1). This is the formula for the nth term of an arithmetic progression.

This is the sum formula n members of an arithmetic progression.

The arithmetic progression is named because in it each term, except for the first, is equal to the arithmetic mean of the two adjacent to it - the previous and the next, indeed,

Geometric progression

A numerical sequence, all members of which are non-zero and each member of which, starting from the second, is obtained from the previous member by multiplying by the same number q, is calledgeometric progression, and the number q is the denominator of a geometric progression. Thus, a geometric progression is a numerical sequence (given recursively by the relations

B, = q (n = 2, 3, 4…; b and q are given numbers).

Example 1. 2, 6, 18, 54, ... - increasing geometric progression

2, q = 3.

Example 2. 2, -2, 2, -2, ... is a geometric progression= 2, q = –1.

One of the obvious properties of a geometric progression is that if a sequence is a geometric progression, then the sequence of squares, i.e.; ;…-

is a geometric progression whose first term is equal to, and the denominator is.

The formula for the nth member of a geometric progression is:

The formula for the sum of n members of a geometric progression:

characteristic propertygeometric progression: a number sequence is a geometric progression if and only if the square of each of its terms, except the first (and the last in the case of a finite sequence), is equal to the product of the previous and subsequent terms,

Conclusion

Numerical sequences have been studied by many scientists for many centuries.The first progression problems that have come down to us are connected with the demands of economic life and social practice, such as the distribution of products, the division of inheritance, and so on. They are one of the key concepts of mathematics. In my work, I tried to reflect the basic concepts associated with numerical sequences, how to set them, properties, and considered some of them. Separately, progressions (arithmetic and geometric) were considered, and the basic concepts associated with them were described.

Bibliography

1. A.G. Mordkovich, Algebra, Grade 10, textbook, 2012
2. A.G. Mordkovich, Algebra, grade 9, textbook, 2012
3. Great student guide. Moscow, "Drofa", 2001
4. G.I. Glaser, History of Mathematics in the School,

M.: Enlightenment, 1964.

1. "Mathematics at school", magazine, 2002.
2. Educational online services Webmath.ru
3. Universal popular science online encyclopedia "Krugosvet"

Vida y= f(x), x O N, where N is the set of natural numbers (or the function of a natural argument), denoted y=f(n) or y 1 ,y 2 ,…, y n,…. Values y 1 ,y 2 ,y 3 ,… are called respectively the first, second, third, ... members of the sequence.

For example, for the function y= n 2 can be written:

y 1 = 1 2 = 1;

y 2 = 2 2 = 4;

y 3 = 3 2 = 9;…y n = n 2 ;…

Methods for setting sequences. Sequences can be specified in various ways, among which three are especially important: analytical, descriptive, and recurrent.

1. A sequence is given analytically if its formula is given n-th member:

y n=f(n).

Example. y n= 2n- 1 – sequence of odd numbers: 1, 3, 5, 7, 9, ...

2. Descriptive the way to specify 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." This means that we are talking about a 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 way of specifying the sequence in this example, it is difficult to answer what, say, the 1000th element of the sequence is equal to.

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

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

Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; ….

It can be seen that the sequence obtained in this example can also be specified analytically: y n= 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 because it has a number of 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.

Properties of numerical sequences.

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

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

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 > … .

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

Example 1 y 1 = 1; y n= n 2 is an increasing sequence.

Thus, the following theorem is true (a characteristic property of an arithmetic progression). A numerical sequence is arithmetic if and only if each of its members, except for the first (and last in the case of a finite sequence), is equal to the arithmetic mean of the previous and subsequent members.

Example. At what value x number 3 x + 2, 5x– 4 and 11 x+ 12 form a finite arithmetic progression?

According to the characteristic property, the given expressions must satisfy the relation

5x – 4 = ((3x + 2) + (11x + 12))/2.

Solving this equation gives x= –5,5. With this value x given expressions 3 x + 2, 5x– 4 and 11 x+ 12 take, respectively, the values ​​-14.5, –31,5, –48,5. This is an arithmetic progression, its difference is -17.

Geometric progression.

A numerical sequence, all members of which are nonzero and each member of which, starting from the second, is obtained from the previous member by multiplying by the same number q, is called a geometric progression, and the number q- the denominator of a geometric progression.

Thus, a geometric progression is a numerical sequence ( b n) given recursively by the relations

b 1 = b, b n = b n –1 q (n = 2, 3, 4…).

(b and q- given numbers, b ≠ 0, q ≠ 0).

Example 1. 2, 6, 18, 54, ... - increasing geometric progression b = 2, q = 3.

Example 2. 2, -2, 2, -2, ... – geometric progression b= 2,q= –1.

Example 3. 8, 8, 8, 8, … – geometric progression b= 8, q= 1.

A geometric progression is an increasing sequence if b 1 > 0, q> 1, and decreasing if b 1 > 0, 0q

One of the obvious properties of a geometric progression is that if a sequence is a geometric progression, then the sequence of squares, i.e.

b 1 2 , b 2 2 , b 3 2 , …, b n 2,… is a geometric progression whose first term is equal to b 1 2 , and the denominator is q 2 .

Formula n- th term of a geometric progression has the form

b n= b 1 q n– 1 .

You can get the formula for the sum of terms of a finite geometric progression.

Let there be a finite geometric progression

b 1 ,b 2 ,b 3 , …, b n

let S n - the sum of its members, i.e.

S n= b 1 + b 2 + b 3 + … +b n.

It is accepted that q No. 1. To determine S n an artificial trick is applied: some geometric transformations of the expression are performed S n q.

S n q = (b 1 + b 2 + b 3 + … + b n –1 + b n)q = b 2 + b 3 + b 4 + …+ b n+ b n q = S n+ b n q– b 1 .

In this way, S n q= S n +b n q – b 1 and hence

This is the formula with umma n members of a geometric progression for the case when q≠ 1.

At q= 1 formula can not be derived separately, it is obvious that in this case S n= a 1 n.

It is called a geometric progression because in it each term, except for the first, is equal to the geometric mean of the previous and subsequent terms. Indeed, since

b n = b n- 1 q;

bn = bn+ 1 /q,

Consequently, b n 2= b n– 1 bn+ 1 and the following theorem is true (a characteristic property of a geometric progression):

a numerical sequence is a geometric progression if and only if the square of each of its terms, except the first (and the last in the case of a finite sequence), is equal to the product of the previous and subsequent terms.

Sequence limit.

Let there be a sequence ( c n} = {1/n}. This sequence is called harmonic, since each of its members, starting from the second, is the harmonic mean between the previous and subsequent members. Geometric mean of numbers a and b there is a number

Otherwise, the sequence is called divergent.

Based on this definition, one can, for example, prove the existence of a limit A=0 for the harmonic sequence ( c n} = {1/n). Let ε be an arbitrarily small positive number. We consider the difference

Is there such N that for everyone n≥ N inequality 1 /N? If taken as N any natural number greater than 1/ε , then for all n ≥ N inequality 1 /n ≤ 1/N ε , Q.E.D.

It is sometimes very difficult to prove the existence of a limit for a particular sequence. The most common sequences are well studied and are listed in reference books. There are important theorems that make it possible to conclude that a given sequence has a limit (and even calculate it) based on already studied sequences.

Theorem 1. If a sequence has a limit, then it is bounded.

Theorem 2. If a sequence is monotone and bounded, then it has a limit.

Theorem 3. If the sequence ( a n} has a limit A, then the sequences ( ca n}, {a n+ c) and (| a n|} have limits cA, A +c, |A| respectively (here c is an arbitrary number).

Theorem 4. If sequences ( a n} and ( b n) have limits equal to A and B pa n + qb n) has a limit pA+ qB.

Theorem 5. If sequences ( a n) and ( b n) have limits equal to A and B respectively, then the sequence ( a n b n) has a limit AB.

Theorem 6. If sequences ( a n} and ( b n) have limits equal to A and B respectively, and in addition b n ≠ 0 and B≠ 0, then the sequence ( a n / b n) has a limit A/B.

Anna Chugainova