What is a sequence of natural numbers. Some types of sequences

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

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

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

briefly designated and called numerical sequence. Magnitude a n is called a common member of a number sequence. Usually the number sequence is given by some formula a n = f(n) allowing 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 define a numerical sequence using 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 differ at least in their numbers, of which there are infinitely many.

A number 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(decreasing), if for any n  N
Such sequences are called strictly monotonous.

Sometimes it is convenient to use not all natural numbers as 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 as another number to the set of natural numbers). In such cases, when 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 number 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, … – a numerical sequence of approximate values with increasing accuracy.

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

Example 4 . Write the first 5 terms of a number sequence using its common term
. To calculate a 1 is needed in the formula for the general 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:

Test 7 .
is:

Test 8 . Common member of the sequence
is:

Number sequence limit

Consider a number sequence whose common term approaches some number A when the serial number increases n. In this case, the number sequence is said to have a limit. This concept has a more strict definition.

Number A called the limit of a 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 A there is a limit to a number sequence if its common term approaches without limit A 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 ( A – , A+ ). A sequence having a limit is called convergent; otherwise – divergent.

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

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

The sequence is called limited, if such a number exists M, What
for everyone n. Every convergent sequence is bounded. Every monotonic 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 approximately equal to 2.718 28.

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

1) convergent;

2) divergent;

3) limited;

Test 10 . Subsequence
is:

1) convergent;

2) divergent;

3) limited;

4) arithmetic progression;

5) geometric progression.

Test 11 . Subsequence is not:

1) convergent;

2) divergent;

3) limited;

4) harmonic.

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

A natural number is a quantitative characteristic of one unchanging set, however, in practice, the number of objects is constantly changing, for example, the number of livestock on a certain farm. Moreover, the simplest, but also the most important sequence immediately appears in the counting process - this is the sequence of natural numbers: 1, 2, 3, ....

If a change in the number of objects in a certain population is fixed in the form of a certain sequence of natural numbers (members of the sequence), another sequence immediately arises naturally - a sequence of numbers, for example

In this regard, the problem of naming the members of a sequence arises. Designating each member with a special letter is extremely inconvenient for the following reasons. First, the sequence may contain a very large, or even an infinite number of terms. Secondly, different letters hide the fact that the members of the sequence belong to the same population, although changing the number of elements. Finally, in this case the member numbers in the sequence will not be reflected.

These reasons force us to designate the members of the sequence with one letter and distinguish them by index. For example, a sequence consisting of ten terms can be denoted by the letter A: A 1 , A 2 , A 3 , …, A 10. The fact that the sequence is infinite is expressed by the ellipsis, as if extending this sequence indefinitely: A 1 , A 2 , A 3, ... Sometimes the sequence begins to be numbered from scratch: : A 0 , A 1 , A 2 , A 3 , …

Some sequences can be perceived as random sets of numbers, since the law of formation of the sequence members is unknown, or even absent. However, special attention is drawn to sequences for which such a law is known.

To indicate the law of formation of sequence members, two methods are most often used. The first of them is as follows. The first term is specified, and then the method is specified according to which the next one is obtained using the last, already known term. To write a law, a sequence member with an unspecified number is used, for example, and k and the next member and k +1, after which the formula connecting them is written.

The most famous and important examples are arithmetic and geometric progressions. Arithmetic progression is defined by the formula and k +1 = and k + r(or and k +1 = and k – r). The terms of an arithmetic progression either increase uniformly (like a ladder) or decrease uniformly (also like a ladder). Magnitude r is called the progression difference because and k +1 – and k = r. Examples of arithmetic progressions with natural terms are

a) natural numbers ( a 1 = 1 ;and k +1 = and k + 1);

b) an infinite sequence 1, 3, 5, 7, … ( a 1 = 1 ;and k +1 = and k + 2);

c) the final sequence 15, 12, 9, 6, 3 ( a 1 = 15 ;and k +1 = and k–3 ).

Geometric progression is given by the formula b k +1 = b k ∙q. Magnitude q is called the denominator of a geometric progression because b k +1:b k = q. Geometric progressions with natural terms and a denominator exceeding one grow and grow quickly, even like an avalanche. Examples of geometric progressions with natural terms are

a) an infinite sequence 1, 2, 4, 8, … ( b 1 = 1 ;b k +1 = b k ∙2);

b) infinite sequence 3, 12, 48, 192, 768,… ( b 1 = 3 ;b k +1 = b k ∙4).

The second way to indicate the law for determining the terms of a sequence is to indicate a formula that allows you to calculate a sequence member with an unspecified number (common term), for example, and k, using the number k.

The terms of arithmetic and geometric progressions can also be calculated in this way. Since the arithmetic progression is defined by the formula and k +1 = and k + r, it is easy to understand how the member is expressed and k using the number k:

a 1– determined arbitrarily;

a 2 = a 1 + r= a 1 + 1∙r;

a 3 = a 2 + r = a 1 + r + r = a 1 + 2∙r;

a 4 = a 3 + r = a 1 + 2∙r + r = a 1 + 3∙r;

…………………………………

and k = a 1 + (k–1)∙r– final formula.

For a geometric progression, the formula for the general term is derived in a similar way: b k = b 1 ∙ q k –1 .

In addition to arithmetic and geometric progressions, other sequences that have a special character of change can be determined in the same way. As an example, we give a sequence of squares of natural numbers: s k = k 2: 1 2 = 1, 2 2 = 4, 3 2 = 9, 4 2 = 16, 5 2 = 25…

There are more complex ways of forming sequences, for example, one is built with the help of another. Of particular importance for arithmetic is the geometric progression determined by the parameters b 1 = 1, q= 10, that is, the sequence of powers of ten: 1 = 10 0, 10 = 10 1, 10 2, 10 3, ..., 10 k, ... It is used to represent natural numbers in the positional number system. Moreover, for each natural number n a sequence appears consisting of numbers with which the given number is written: a n a n – 1 ... a 2 a 1 a 0. Number and k indicates how many terms of type 10 k contains a number n.

The concept of sequence leads to the most important concepts of quantity and function for mathematics. A quantity is a changing numerical characteristic of an object or phenomenon. Its change is perceived as a sequence of numbers. The existence of a relationship between the terms themselves and their numbers, as well as its expression using formulas, leads closely to the concept of a function.

10. Decimal number system.

The most important mathematical discovery, which is used by almost every member of a fairly developed society, is the positional number system. It made it possible to solve the main problem of counting, which is the ability to name more and more new numbers, using notations (digits) only for the first few numbers.

The positional number system is traditionally associated with the number ten, but other systems, for example, binary, can be built on the same principles. When constructing a decimal positional number system, ten Arabic numerals are introduced: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. With their help, a number can be written that expresses the number of objects of any finite set. For this purpose, a special algorithm is used, that is, a clearly defined sequence of elementary actions.

The items being counted are combined into groups of ten, which corresponds to division by ten with a remainder. As a result, two sets are formed - ones and tens. The tens are again grouped by tens into hundreds. It is clear that the number of tens (we denote it by a 1) is necessarily less than ten, and, therefore, a 1 can be indicated by a number. Then hundreds are grouped into thousands, thousands into tens of thousands, etc. until all items are grouped. The construction of the number is completed by writing the resulting numbers from left to right from large indices to smaller ones. Digital and k correspond to the number of groups of objects of 10 k. The final record of a number consists of a finite sequence of digits a n a n – 1 ... a 2 a 1 a 0. The corresponding number is equal to the expression

а n ·10 n + а n – 1 ·10 n – 1 + … + а 2 ·10 2 + а 1 ·10 1 + а 0 ·10 0.

The word “positional” in the name of the number system is due to the fact that a number changes its meaning depending on its position in the notation of the number. The last digit specifies the number of units, the penultimate digit specifies the number of tens, etc.

Note that the algorithm for obtaining a record of numbers in a number system with any base N: consists of sequential grouping of objects according to N things. When writing numbers you must use N numbers

Subsequence

Subsequence- This kit elements of some set:

for each natural number you can specify an element of a given set;
this number is the number of the element and indicates the position of this element in the sequence;
For any element (member) of a sequence, you can specify the next element of the sequence.

So the sequence turns out to be the result consistent selection of elements of a given set. And, if any set of elements is finite, and we talk about a sample of finite volume, then the sequence turns out to be a sample of infinite volume.

A sequence is by its nature a mapping, so it should not be confused with a set that “runs through” the sequence.

In mathematics, many different sequences are considered:

time series of both numerical and non-numerical nature;
sequences of elements of metric space
sequences of functional space elements
sequences of states of control systems and machines.

The purpose of studying all possible sequences is to search for patterns, predict future states and generate sequences.

Definition

Let a certain set of elements of arbitrary nature be given. | Any mapping from a set of natural numbers to a given set is called sequence(elements of the set).

The image of a natural number, namely, the element, is called - th member or sequence element, and the ordinal number of a member of the sequence is its index.

Related definitions

If we take an increasing sequence of natural numbers, then it can be considered as a sequence of indices of some sequence: if we take the elements of the original sequence with the corresponding indices (taken from the increasing sequence of natural numbers), then we can again get a sequence called subsequence given sequence.

Comments

In mathematical analysis, an important concept is the limit of a number sequence.

Designations

Sequences of the form

It is customary to write compactly using parentheses:

Curly braces are sometimes used:

Allowing some freedom of speech, we can also consider finite sequences of the form

which represent the image of the initial segment of a sequence of natural numbers.

Classes of natural numbers.

Any natural number can be written using 10 Arabic numerals:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

To read natural numbers, they are divided, starting from the right, into groups of 3 digits each. 3 first the numbers on the right are the class of units, the next 3 are the class of thousands, then the classes of millions, billions andso on. Each of the class digits is called itsdischarge.

Comparison of natural numbers.

Of 2 natural numbers, the smaller is the number that is called earlier when counting. For example, number 7 less 11 (write like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .

Table of digits and classes of numbers.


1st class unit	1st digit of the unit 2nd digit tens 3rd place hundreds
2nd class thousand	1st digit of unit of thousands 2nd digit tens of thousands 3rd category hundreds of thousands
3rd class millions	1st digit of unit of millions 2nd category tens of millions 3rd category hundreds of millions
4th class billions	1st digit of unit of billions 2nd category tens of billions 3rd category hundreds of billions

Numbers from 5th grade and above are considered large numbers. Units of the 5th class are trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - eptillions.

Basic properties of natural numbers.

Commutativity of addition . a + b = b + a
Commutativity of multiplication. ab = ba
Associativity of addition. (a + b) + c = a + (b + c)
Associativity of multiplication.
Distributivity of multiplication relative to addition:

Operations on natural numbers.

4. Division of natural numbers is the inverse operation of multiplication.

If b ∙ c = a, That

Formulas for division:

a: 1 = a

a: a = 1, a ≠ 0

0: a = 0, a ≠ 0

(A∙ b) : c = (a:c) ∙ b

(A∙ b) : c = (b:c) ∙ a

Numerical expressions and numerical equalities.

A notation where numbers are connected by action signs is numerical expression.

For example, 10∙3+4; (60-2∙5):10.

Records where 2 numeric expressions are combined with an equal sign are numerical equalities. Equality has left and right sides.

The order of performing arithmetic operations.

Adding and subtracting numbers are operations of the first degree, while multiplication and division are operations of the second degree.

When a numerical expression consists of actions of only one degree, they are performed sequentially from left to right.

When expressions consist of actions of only the first and second degrees, then the actions are performed first second degree, and then - actions of the first degree.

When there are parentheses in an expression, the actions in the parentheses are performed first.

For example, 36:(10-4)+3∙5= 36:6+15 = 6+15 = 21.