Vida y= f(x), x O N, where N is the set of natural numbers (or a 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 specify 1–2 initial members of the sequence.

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 only natural numbers 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 of whose members 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 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, 0 q

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.

The geometric progression is named because in it each term except 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

Before we start to decide arithmetic progression problems, consider what a number sequence is, since an arithmetic progression is a special case of a number sequence.

A numerical sequence is a numerical set, each element of which has its own serial number. The elements of this set are called members of the sequence. The ordinal number of a sequence element is indicated by an index:

The first element of the sequence;

The fifth element of the sequence;

- "nth" element of the sequence, i.e. the element "standing in the queue" at number n.

There is a dependency between the value of a sequence element and its ordinal number. Therefore, we can consider a sequence as a function whose argument is the ordinal number of an element of the sequence. In other words, one can say that the sequence is a function of the natural argument:

The sequence can be specified in three ways:

1 . The sequence can be specified using a table. In this case, we simply set the value of each member of the sequence.

For example, Someone decided to do personal time management, and to begin with, to calculate how much time he spends on VKontakte during the week. By writing the time in a table, he will get a sequence consisting of seven elements:

The first line of the table contains the number of the day of the week, the second - the time in minutes. We see that, that is, on Monday Someone spent 125 minutes on VKontakte, that is, on Thursday - 248 minutes, and, that is, on Friday, only 15.

2 . The sequence can be specified using the nth member formula.

In this case, the dependence of the value of a sequence element on its number is expressed directly as a formula.

For example, if , then

To find the value of a sequence element with a given number, we substitute the element number into the formula for the nth member.

We do the same if we need to find the value of a function if the value of the argument is known. We substitute the value of the argument instead in the equation of the function:

If, for example, , then

Once again, I note that in a sequence, in contrast to an arbitrary numeric function, only a natural number can be an argument.

3 . The sequence can be specified using a formula that expresses the dependence of the value of the member of the sequence with number n on the value of the previous members. In this case, it is not enough for us to know only the number of a sequence member in order to find its value. We need to specify the first member or first few members of the sequence.

For example, consider the sequence ,

We can find the values ​​of the members of a sequence in sequence, starting from the third:

That is, each time to find the value of the nth member of the sequence, we return to the previous two. This way of sequencing is called recurrent, from the Latin word recurro- come back.

Now we can define an arithmetic progression. An arithmetic progression is a simple special case of a numerical sequence.

Arithmetic progression is called a numerical sequence, each member of which, starting from the second, is equal to the previous one, added with the same number.

The number is called the difference of an arithmetic progression. The difference of an arithmetic progression can be positive, negative, or zero.

If title="(!LANG:d>0">, то каждый член арифметической прогрессии больше предыдущего, и прогрессия является !} increasing.

For example, 2; 5; eight; eleven;...

If , then each term of the arithmetic progression is less than the previous one, and the progression is waning.

For example, 2; -one; -four; -7;...

If , then all members of the progression are equal to the same number, and the progression is stationary.

For example, 2;2;2;2;...

The main property of an arithmetic progression:

Let's look at the picture.

We see that

, and at the same time

Adding these two equalities, we get:

.

Divide both sides of the equation by 2:

So, each member of the arithmetic progression, starting from the second, is equal to the arithmetic mean of two neighboring ones:

Moreover, since

, and at the same time

, then

, and hence

Each member of the arithmetic progression starting with title="(!LANG:k>l">, равен среднему арифметическому двух равноотстоящих. !}

th member formula.

We see that for the members of the arithmetic progression, the following relations hold:

and finally

We got formula of the nth term.

IMPORTANT! Any member of an arithmetic progression can be expressed in terms of and . Knowing the first term and the difference of an arithmetic progression, you can find any of its members.

The sum of n members of an arithmetic progression.

In an arbitrary arithmetic progression, the sums of terms equally spaced from the extreme ones are equal to each other:

Consider an arithmetic progression with n members. Let the sum of n members of this progression be equal to .

Arrange the terms of the progression first in ascending order of numbers, and then in descending order:

Let's pair it up:

The sum in each parenthesis is , the number of pairs is n.

We get:

So, the sum of n members of an arithmetic progression can be found using the formulas:

Consider solving arithmetic progression problems.

1 . The sequence is given by the formula of the nth member: . Prove that this sequence is an arithmetic progression.

Let us prove that the difference between two adjacent members of the sequence is equal to the same number.

We have obtained that the difference of two adjacent members of the sequence does not depend on their number and is a constant. Therefore, by definition, this sequence is an arithmetic progression.

2 . Given an arithmetic progression -31; -27;...

a) Find the 31 terms of the progression.

b) Determine if the number 41 is included in this progression.

a) We see that ;

Let's write down the formula for the nth term for our progression.

In general

In our case , that's why

We get:

b) Suppose the number 41 is a member of the sequence. Let's find his number. To do this, we solve the equation:

We got a natural value of n, therefore, yes, the number 41 is a member of the progression. If the found value of n were not a natural number, then we would answer that the number 41 is NOT a member of the progression.

3 . a) Between the numbers 2 and 8, insert 4 numbers so that they, together with the given numbers, form an arithmetic progression.

b) Find the sum of the terms of the resulting progression.

a) Let's insert four numbers between the numbers 2 and 8:

We got an arithmetic progression, in which there are 6 members.

Let's find the difference of this progression. To do this, we use the formula for the nth term:

Now it's easy to find the values ​​of the numbers:

3,2; 4,4; 5,6; 6,8

b)

Answer: a) yes; b) 30

4. The truck transports a batch of crushed stone weighing 240 tons, daily increasing the transportation rate by the same number of tons. It is known that 2 tons of rubble were transported on the first day. Determine how many tons of crushed stone were transported on the twelfth day if all the work was completed in 15 days.

According to the condition of the problem, the amount of crushed stone that the truck transports increases every day by the same number. Therefore, we are dealing with an arithmetic progression.

We formulate this problem in terms of an arithmetic progression.

During the first day, 2 tons of crushed stone were transported: a_1=2.

All work was completed in 15 days: .

The truck transports a batch of crushed stone weighing 240 tons:

We need to find .

First, let's find the progression difference. Let's use the formula for the sum of n members of the progression.

In our case:

Numeric sequence.
How ?

In this lesson, we will learn a lot of interesting things from the life of members of a large community called Vkontakte number sequences. The topic under consideration refers not only to the course of mathematical analysis, but also touches on the basics discrete mathematics. In addition, the material will be required for the development of other sections of the tower, in particular, during the study number series and functional rows. You can tritely say that this is important, you can say reassuringly that it’s simple, you can say a lot more on-duty phrases, but today is the first, unusually lazy school week, so it’s terribly breaking for me to compose the first paragraph =) I already saved the file in my heart and got ready to sleep, suddenly… the idea of ​​a frank confession lit up the head, which incredibly relieved the soul and pushed for further tapping of the fingers on the keyboard.

Let's digress from summer memories and look into this fascinating and positive world of a new social network:

The concept of a numerical sequence

First, let's think about the word itself: what is a sequence? Consistency is when something is located behind something. For example, the sequence of actions, the sequence of the seasons. Or when someone is located behind someone. For example, a sequence of people in a queue, a sequence of elephants on a path to a watering hole.

Let us immediately clarify the characteristic features of the sequence. Firstly, sequence members are located strictly in a certain order. So, if two people in the queue are swapped, then this will already be another subsequence. Secondly, to each sequence member you can assign a serial number:

It's the same with numbers. Let to each natural value according to some rule mapped to a real number. Then we say that a numerical sequence is given.

Yes, in mathematical problems, in contrast to life situations, the sequence almost always contains infinitely many numbers.

Wherein:
called first member sequences;
– second member sequences;
– third member sequences;
…
– nth or common member sequences;
…

In practice, the sequence is usually given common term formula, for example:
is a sequence of positive even numbers:

Thus, the record uniquely determines all members of the sequence - this is the rule (formula) according to which the natural values numbers are matched. Therefore, the sequence is often briefly denoted by a common member, and other Latin letters can be used instead of "x", for example:

Sequence of positive odd numbers:

Another common sequence:

As, probably, many have noticed, the variable "en" plays the role of a kind of counter.

In fact, we dealt with numerical sequences back in middle school. Let's remember arithmetic progression. I will not rewrite the definition, let's touch on the very essence with a specific example. Let be the first term and step arithmetic progression. Then:
is the second term of this progression;
is the third member of this progression;
- fourth;
- fifth;
…
And, obviously, the nth member is asked recurrent formula

Note : in a recursive formula, each next term is expressed in terms of the previous term or even in terms of a whole set of previous terms.

The resulting formula is of little use in practice - to get, say, to , you need to go through all the previous terms. And in mathematics, a more convenient expression for the nth term of an arithmetic progression is derived: . In our case:

Substitute natural numbers in the formula and check the correctness of the numerical sequence constructed above.

Similar calculations can be made for geometric progression, the nth term of which is given by the formula , where is the first term , and is denominator progressions. In matan assignments, the first term is often equal to one.

progression sets the sequence ;
progression sets the sequence ;
progression sets the sequence ;
progression sets the sequence .

I hope everyone knows that -1 to an odd power is -1, and to an even power is one.

The progression is called infinitely decreasing, if (last two cases).

Let's add two new friends to our list, one of which has just knocked on the monitor matrix:

The sequence in mathematical jargon is called a "flasher":

In this way, sequence members can be repeated. So, in the considered example, the sequence consists of two infinitely alternating numbers.

Does it happen that the sequence consists of the same numbers? Of course. For example, it sets an infinite number of "triples". For aesthetes, there is a case when “en” still formally appears in the formula:

Let's invite a simple girlfriend to dance:

What happens when "en" increases to infinity? Obviously, the terms of the sequence will infinitely close approach zero. This is the limit of this sequence, which is written as follows:

If the limit of a sequence is zero, then it is called infinitesimal.

In the theory of mathematical analysis, it is given strict definition of the sequence limit through the so-called epsilon neighborhood. The next article will be devoted to this definition, but for now let's analyze its meaning:

Let us depict the terms of the sequence and the neighborhood symmetric with respect to zero (limit) on the real line:


Now hold the blue neighborhood with the edges of your palms and start to reduce it, pulling it to the limit (red dot). A number is the limit of a sequence if FOR ANY pre-selected -neighborhood (arbitrarily small) inside it will be infinitely many members of the sequence, and OUTSIDE of it - only final number of members (or none at all). That is, the epsilon neighborhood can be microscopic, and even less, but the “infinite tail” of the sequence must sooner or later fully enter this area.

The sequence is also infinitely small: with the difference that its members do not jump back and forth, but approach the limit exclusively from the right.

Naturally, the limit can be equal to any other finite number, an elementary example:

Here the fraction tends to zero, and accordingly, the limit is equal to "two".

If the sequence there is a finite limit, then it is called converging(in particular, infinitesimal at ). Otherwise - divergent, while two options are possible: either the limit does not exist at all, or it is infinite. In the latter case, the sequence is called infinitely large. Let's gallop through the examples of the first paragraph:

Sequences are infinitely large, as their members move steadily towards "plus infinity":

An arithmetic progression with the first term and a step is also infinitely large:

By the way, any arithmetic progression also diverges, except for the case with a zero step - when infinitely added to a specific number. The limit of such a sequence exists and coincides with the first term.

Sequences have a similar fate:

Any infinitely decreasing geometric progression, as the name implies, infinitely small:

If the denominator is a geometric progression, then the sequence is infinitely largeA:

If, for example, , then there is no limit at all, since the members tirelessly jump either to “plus infinity”, then to “minus infinity”. And common sense and matan's theorems suggest that if something strives somewhere, then this cherished place is unique.

After a little revelation it becomes clear that the flasher is to blame for the unrestrained throwing, which, by the way, diverges on its own.
Indeed, for a sequence it is easy to choose a -neighbourhood, which, say, clamps only the number -1. As a result, an infinite number of sequence members (“plus ones”) will remain outside the given neighborhood. But by definition, the "infinite tail" of the sequence from a certain moment (natural number) must fully enter ANY neighborhood of its limit. Conclusion: there is no limit.

Factorial is infinitely large sequence:

Moreover, it grows by leaps and bounds, so it is a number that has more than 100 digits (digits)! Why exactly 70? It asks for mercy my engineering calculator.

With a control shot, everything is a little more complicated, and we have just come to the practical part of the lecture, in which we will analyze combat examples:

But now it is necessary to be able to solve the limits of functions, at least at the level of two basic lessons: Limits. Solution examples and Remarkable Limits. Because many solution methods will be similar. But, first of all, let's analyze the fundamental differences between the limit of a sequence and the limit of a function:

In the limit of the sequence, the "dynamic" variable "en" can tend to only to "plus infinity"– in the direction of increasing natural numbers .
In the limit of the function, "x" can be directed anywhere - to "plus / minus infinity" or to an arbitrary real number.

Subsequence discrete(discontinuous), that is, it consists of separate isolated members. One, two, three, four, five, the bunny went out for a walk. The argument of the function is characterized by continuity, that is, “x” smoothly, without incident, tends to one or another value. And, accordingly, the values ​​of the function will also continuously approach their limit.

Because of discreteness within the sequences there are their own branded things, such as factorials, flashers, progressions, etc. And now I will try to analyze the limits that are characteristic of sequences.

Let's start with progressions:

Example 1

Find the limit of a sequence

Solution: something similar to an infinitely decreasing geometric progression, but is it really? For clarity, we write out the first few terms:

Since , we are talking about sum members of an infinitely decreasing geometric progression, which is calculated by the formula .

Making a decision:

We use the formula for the sum of an infinitely decreasing geometric progression: . AT this case: - the first term, - the denominator of the progression.

Example 2

Write the first four terms of the sequence and find its limit

This is a do-it-yourself example. To eliminate the uncertainty in the numerator, you will need to apply the formula for the sum of the first terms of an arithmetic progression:
, where is the first and is the nth term of the progression.

Since 'en' always tends to 'plus infinity' within sequences, it's not surprising that indeterminacy is one of the most popular.
And many examples are solved in exactly the same way as the limits of functions!

Or maybe something more complicated like ? Check out Example #3 of the article Limit Solving Methods.

From a formal point of view, the difference will be only in one letter - there is “x”, and here “en”.
The reception is the same - the numerator and denominator must be divided by "en" in the highest degree.

Also, within sequences, uncertainty is quite common. How to solve limits like can be found in Examples No. 11-13 of the same article.

To deal with the limit, refer to Example #7 of the lesson Remarkable Limits(the second remarkable limit is also valid for the discrete case). The solution will again be like a carbon copy with a difference in a single letter.

The following four examples (Nos. 3-6) are also “two-faced”, but in practice, for some reason, they are more typical for the limits of sequences than for the limits of functions:

Example 3

Find the limit of a sequence

Solution: first complete solution, then step by step comments:

(1) In the numerator we use the formula twice.

(2) We give like terms in the numerator.

(3) To eliminate uncertainty, we divide the numerator and denominator by ("en" in the highest degree).

As you can see, nothing complicated.

Example 4

Find the limit of a sequence

This is an example for a do-it-yourself solution, abbreviated multiplication formulas to help.

Within s demonstrative sequences use a similar method of dividing the numerator and denominator:

Example 5

Find the limit of a sequence

Solution let's do it the same way:

A similar theorem is also true, by the way, for functions: the product of a bounded function by an infinitesimal function is an infinitesimal function.

Example 9

Find the limit of a sequence

Hovhannisyan Eva

Numeric sequences. Abstract.

Download:

Preview:

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 terms.

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 found for the first time 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 any instructions on 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 this 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 is bounded from above by unity, and therefore 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 term, starting from the second, is obtained by adding the number 4 to the previous term. 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"

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 specify 1–2 initial members of the sequence.

• 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 numeric sequence to calculate the n-th element, we just need to specify 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 also be considered for sequences.

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.