Methods for setting numerical sequences. Presentation: Numerical sequence concept and types
2. Determine the arithmetic operation, with the help of which the average is obtained from the two extreme numbers, and instead of the * sign, insert the missing number: eight
3. The students solved the task in which it is required to find the missing numbers. They got different answers. Find the rules by which the guys filled in the cells. Task Answer 1 Answer
Definition of a numerical sequence It is said that a numerical sequence is given if, according to some law, a certain number (a member of the sequence) is uniquely assigned to any natural number (place number). In general terms, this correspondence can be represented as follows: y 1, y 2, y 3, y 4, y 5, ..., y n, ... ... n ... The number n is the n-th member of the sequence. The entire sequence is usually denoted (y n).
Analytical way of specifying numerical sequences A sequence is specified analytically if the formula of the nth member is specified. For example, 1) y n= n 2 - analytical assignment of the sequence 1, 4, 9, 16, ... 2) y n= С - constant (stationary) sequence 2) y n= 2 n - analytical assignment of the sequence 2, 4, 8, 16, … Solve 585
Recurrent method of specifying numerical sequences The recurrent method of specifying a sequence consists in specifying a rule that allows calculating the n-th term if its previous terms are known 1) arithmetic progression is given by recursive relations a 1 =a, a n+1 =a n + d 2) geometric progression - b 1 =b, b n+1 =b n * q
Anchoring 591, 592 (a, b) 594, – 614 (a)
Upper Bounded A sequence (y n) is said to be bounded from above if all its members are at most some number. In other words, a sequence (y n) is bounded from above if there exists a number M such that for any n the inequality y n M holds. M is the upper bound of the sequence For example, -1, -4, -9, -16, …, -n 2, …
Bounded from below A sequence (y n) is called bounded from below if all its members are at least some number. In other words, the sequence (y n) is bounded from above if there exists a number m such that for any n the inequality y n m holds. m is the lower bound of the sequence For example, 1, 4, 9, 16, …, n 2, …
Boundedness of a Sequence A sequence (y n) is called bounded if it is possible to specify two numbers A and B between which all members of the sequence lie. The inequality Ay n B A is the lower bound, B is the upper bound For example, 1 is the upper bound, 0 is the lower bound
Decreasing sequence A sequence is called decreasing if each member is less than the previous one: y 1 > y 2 > y 3 > y 4 > y 5 > ... > y n > ... For example, y 2 > y 3 > y 4 > y 5 > … > y n > … For example, "> y 2 > y 3 > y 4 > y 5 > … > y n > … For example,"> y 2 > y 3 > y 4 > y 5 > … > y n > … For example," title="(!LANG:Descending sequence A sequence is called decreasing if each of its members is less than the previous one: y 1 > y 2 > y 3 > y 4 > y 5 > … > y n > … For example,">
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Verification work Option 1Option 2 1. The numerical sequence is given by the formula a) Calculate the first four terms of this sequence b) Is the number a member of the sequence? b) Is the number 12.25 a member of the sequence? 2. Formulate the th term of the sequence 2, 5, 10, 17, 26,…1, 2, 4, 8, 16,…
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Slides captions:
Number Sequences
Names of months Grades at school Bank account number Houses on the street Sequences make up elements of nature that can be numbered Days of the week
Find patterns and show them with an arrow: 1; four; 7; ten; 13; … In ascending order positive odd numbers 10; 19; 37; 73; 145; ... In descending order, proper fractions with a numerator equal to 1 6; eight; 16; eighteen; 36; … In ascending order, positive multiples of 5 ½; 1/3; ¼; 1/5; 1/6; 3x magnification Alternate 2x magnification and 2x magnification 1; 3; 5; 7; 9; … 5; ten; fifteen; twenty; 25; ... Increase by 2 times and decrease by 1 CHECK YOURSELF
Definition of a numerical sequence A function of the form y \u003d f (x), x belongs to N, is called a function of a natural argument or a numerical sequence and is denoted y \u003d f (n) or y 1, y 2, y 3, ..., y n, ... (Values y 1, y 2, y 3, ... are respectively called the first, second, third (etc.) members of the sequence.In the symbol y n, the number n is called an index that characterizes the ordinal number of one or another member of the sequence (y n)).
Methods for specifying sequences Verbal Recurrent Analytical
Analytical assignment of a numerical sequence If the formula of its n-th member is indicated y n = f (n) For example: Х n =3* n+2 X 5 =3 *5+2=17; X 45 \u003d 3 * 45 + 2 \u003d 137 For example: y n \u003d C C, C, C, ... (stationary)
The sequences are given by the formulas: a n =(-1) n n 2 a n =n 4 a n =n+4 a n =-n- 2 a n =2 n -5 a n =3 n -1 2. Specify what numbers are the members of these sequences Positive and Positive Negative Negative Complete the following tasks: Fill in the missing members of the sequence: 1; ___; 81; ___; 625; … 5; ___; ___; ___; 9; …___; ___; 3; eleven; ___; -one; four; ___; ___; -25; …___; -four ; ___; ___; -7; … 2; eight; ___; ___; ___; … 16 256 6 7 8 -3 -1 27 -9 16 -3 -5 -6 26 80 242 CHECK YOURSELF
Consolidation of the studied material No. 15.1 and 15.2 orally. No. 15.4 on the board and in notebooks. No. 15.10 and 15.11 orally. No. 15.12(c, d) and 15.13 (c, d) with commentary on the spot. No. 15.15 (c, d), 15.16 (c, d), 15.17 (c, d), 15.38 (a, c) on the board and in notebooks.
Lesson Summary: Homework: § 15, pp. 136-139; No. 15.12(a, b), 15.13(a, b), 15.15(a, b), 15.38(b, d).
Thank you for your attention!
On the topic: methodological developments, presentations and notes
Presentation. Sequence of filling of energy levels and sublevels in ChE atoms of small periods
This presentation may be useful as an illustration in the study of the structure of the atom. The presentation shows the sequence of filling energy levels and sublevels in the atoms of chemical e...
Introduction…………………………………………………………………………………3
1.Theoretical part……………………………………………………………….4
Basic concepts and terms…………………………………………………....4
1.1 Types of sequences……………………………………………………...6
1.1.1.Limited and unlimited number sequences…..6
1.1.2.Monotonicity of sequences……………………………………6
1.1.3.Infinitesimal and infinitesimal sequences…….7
1.1.4. Properties of infinitesimal sequences…………………8
1.1.5 Convergent and divergent sequences and their properties..…9
1.2 Sequence Limit…………………………………………………….11
1.2.1.Theorems about the limits of sequences………………………………………………………………15
1.3.Arithmetic progression…………………………………………………………17
1.3.1. Properties of an arithmetic progression……………………………………..17
1.4Geometric progression……………………………………………………..19
1.4.1. Properties of a geometric progression……………………………………….19
1.5. Fibonacci numbers………………………………………………………………..21
1.5.1 Connection of Fibonacci numbers with other areas of knowledge…………………….22
1.5.2. Using a series of Fibonacci numbers to describe animate and inanimate nature……………………………………………………………………………….23
2. Own research…………………………………………………….28
Conclusion………………………………………………………………………….30
List of used literature…………………………………………....31
Introduction.
Number sequences are a very interesting and informative topic. This topic is found in assignments increased complexity, which are offered to students by the authors of didactic materials, in the problems of mathematical Olympiads, entrance exams to higher Educational establishments and on the exam. I am interested to know the connection of mathematical sequences with other fields of knowledge.
Target research work: Expand knowledge of the number sequence.
1. Consider the sequence;
2. Consider its properties;
3. Consider the analytical task of the sequence;
4. Demonstrate its role in the development of other areas of knowledge.
5. Demonstrate the use of a series of Fibonacci numbers to describe animate and inanimate nature.
1. Theoretical part.
Basic concepts and terms.
Definition. A numerical sequence is a function of the form y = f(x), x О N, where N is a set natural numbers(or a function of a natural argument), denoted y = f(n) or y1, y2,…, yn,…. The values y1, y2, y3,… are called respectively the first, second, third, … members of the sequence.
The number a is called the limit of the sequence x \u003d (xn) if for an arbitrary pre-specified arbitrarily small positive number ε there is such a natural number N that for all n>< ε.
A sequence (yn) is called increasing if each of its members (except the first) is greater than the previous one:
y1< y2 < y3 < … < yn < yn+1 < ….
A sequence (yn) is called decreasing if each of its members (except the first) is less than the previous one:
y1 > y2 > y3 > … > yn > yn+1 > … .
Increasing and decreasing sequences are united by a common term - monotonic sequences.
A sequence is called periodic if there exists a natural number T such that, starting from some n, the equality yn = yn+T holds. The number T is called the period length.
An arithmetic progression is a sequence (an), each member of which, starting from the second, is equal to the sum of the previous member and the same number d, is called an arithmetic progression, and the number d is called the difference of an arithmetic progression.
Thus, an arithmetic progression is a numerical sequence (an) given recursively by the relations
a1 = a, an = an–1 + d (n = 2, 3, 4, …)
A geometric progression is a sequence in which all 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.
Thus, a geometric progression is a numerical sequence (bn) given recursively by the relations
b1 = b, bn = bn–1 q (n = 2, 3, 4…).
1.1 Types of sequences.
1.1.1 Bounded and unbounded sequences.
A sequence (bn) is said to be bounded from above if there exists a number M such that for any number n the inequality bn≤ M is satisfied;
A sequence (bn) is said to be bounded from below if there exists a number M such that for any number n the inequality bn≥ M is satisfied;
For example:
1.1.2 Monotonicity of sequences.
A sequence (bn) is called nonincreasing (nondecreasing) if for any number n the inequality bn≥ bn+1 (bn ≤bn+1) is true;
A sequence (bn) is called decreasing (increasing) if for any number n the inequality bn > bn+1 (bn Decreasing and increasing sequences are called strictly monotonic, non-increasing - monotonic in a broad sense. Sequences bounded both above and below are called bounded. The sequence of all these types is called monotonic. 1.1.3 Infinitely large and small sequences. An infinitesimal sequence is a numerical function or sequence that tends to zero. A sequence an is called infinitesimal if A function is called infinitesimal in a neighborhood of the point x0 if ℓimx→x0 f(x)=0. A function is called infinitesimal at infinity if ℓimx→.+∞ f(x)=0 or ℓimx→-∞ f(x)=0 Also infinitesimal is a function that is the difference between a function and its limit, that is, if ℓimx→.+∞ f(x)=а, then f(x) − a = α(x), ℓimx→.+∞ f(( x)-a)=0. An infinitely large sequence is a numerical function or sequence that tends to infinity. A sequence an is called infinitely large if ℓimn→0 an=∞. A function is called infinite in a neighborhood of a point x0 if ℓimx→x0 f(x)= ∞. A function is said to be infinitely large at infinity if ℓimx→.+∞ f(x)= ∞ or ℓimx→-∞ f(x)= ∞ . 1.1.4 Properties of infinitesimal sequences. The sum of two infinitesimal sequences is itself also an infinitesimal sequence. The difference of two infinitesimal sequences is itself also an infinitesimal sequence. The algebraic sum of any finite number of infinitesimal sequences is itself also an infinitesimal sequence. The product of a bounded sequence and an infinitesimal sequence is an infinitesimal sequence. The product of any finite number of infinitesimal sequences is an infinitesimal sequence. Any infinitesimal sequence is bounded. If the stationary sequence is infinitely small, then all its elements, starting from some, are equal to zero. If the entire infinitesimal sequence consists of the same elements, then these elements are zeros. If (xn) is an infinitely large sequence containing no zero terms, then there is a sequence (1/xn) that is infinitesimal. If, however, (xn) contains zero elements, then the sequence (1/xn) can still be defined starting from some number n, and will still be infinitesimal. If (an) is an infinitesimal sequence containing no zero terms, then there is a sequence (1/an) that is infinitely large. If, however, (an) contains zero elements, then the sequence (1/an) can still be defined starting from some number n, and will still be infinitely large. 1.1.5 Convergent and divergent sequences and their properties. A convergent sequence is a sequence of elements of the set X that has a limit in this set. A divergent sequence is a sequence that is not convergent. Every infinitesimal sequence is convergent. Its limit is zero. Removing any finite number of elements from an infinite sequence does not affect either the convergence or the limit of that sequence. Any convergent sequence is bounded. However, not every bounded sequence converges. If the sequence (xn) converges, but is not infinitely small, then, starting from some number, the sequence (1/xn) is defined, which is bounded. The sum of convergent sequences is also a convergent sequence. The difference of convergent sequences is also a convergent sequence. The product of convergent sequences is also a convergent sequence. The quotient of two convergent sequences is defined starting from some element, unless the second sequence is infinitesimal. If the quotient of two convergent sequences is defined, then it is a convergent sequence. If a convergent sequence is bounded below, then none of its lower bounds exceeds its limit. If a convergent sequence is bounded from above, then its limit does not exceed any of its upper bounds. If for any number the terms of one convergent sequence do not exceed the terms of another convergent sequence, then the limit of the first sequence also does not exceed the limit of the second. If all elements of a certain sequence, starting from a certain number, lie on the segment between the corresponding elements of two other sequences converging to the same limit, then this sequence also converges to the same limit. Example. Prove that the sequence (xn)=((2n+1)/n) converges to the number 2. We have |xn-2|=|((2n+1)/n)-2|= 1/n. for any α>0, m belongs to N such that 1/m<α. Тогда n>m the inequality 1/m<α и, следовательно, |xn-1|<α; т.е. ℓimn→∞ xn=2. 1.2 Sequence limit. A number a is called the limit of a sequence x = (xn ) if, for an arbitrary preassigned arbitrarily small positive number ε, there exists a natural number N such that, for all n>N, the inequality |xn - a|< ε. If the number a is the limit of the sequence x \u003d (xn), then they say that xn tends to a, and write. To formulate this definition in geometric terms, we introduce the following notion. A neighborhood of a point x0 is an arbitrary interval (a, b) containing this point inside itself. The neighborhood of the point x0 is often considered, for which x0 is the middle, then x0 is called the center of the neighborhood, and the value (b–a)/2 is called the radius of the neighborhood. So, let's find out what the concept of the limit of a numerical sequence means geometrically. To do this, we write the last inequality from the definition in the form This inequality means that all elements of the sequence with numbers n>N must lie in the interval (a – ε; a + ε). Therefore, the constant number a is the limit of the numerical sequence (xn), if for any small neighborhood centered at a point a of radius ε (ε is a neighborhood of a point a) there is such an element of the sequence with number N that all subsequent elements with numbers n>N will be located within this neighborhood. 1. Let the variable x sequentially take the values Let us prove that the limit of this numerical sequence is equal to 1. Take an arbitrary positive number ε. We need to find a natural number N such that for all n>N the inequality |xn - 1|< ε. Действительно, т.к. then to satisfy the relation |xn - a|< ε достаточно, чтобы Therefore, taking as N any natural number that satisfies the inequality, we get what we need. So if we take, for example, then, setting N=6, for all n>6 we have 2. Using the definition of the limit of a numerical sequence, prove that Take an arbitrary ε > 0. Consider Then, if or, i.e. . Therefore, we choose any natural number that satisfies the inequality Remark 1. Obviously, if all elements of a numerical sequence take the same constant value xn = c, then the limit of this sequence will be equal to the constant itself. Indeed, for any ε, the inequality |xn - c| = |c - c| = 0< ε. Remark 2. It follows from the definition of a limit that a sequence cannot have two limits. Indeed, suppose that xn → a and simultaneously xn → b. Take any and mark the neighborhoods of the points a and b of radius ε. Then, by the definition of the limit, all elements of the sequence, starting from some, must be located both in the neighborhood of the point a and in the neighborhood of the point b, which is impossible. Remark 3. One should not think that every numerical sequence has a limit. Let, for example, the variable takes the values It is easy to see that this sequence does not tend to any limit. Prove that ℓimn→∞qⁿ=0 for |q|< 1. Proof: one). If q=0, then the equality is obvious. Let α> 0 be arbitrary and 0<|q|<1. тогда пользуясь неравенством Бернулли, получим 1/|q|= (1+(1/|q|-1))ⁿ > 1+n(1/|q|-1)> n(1/|q|-1) |q|ⁿ=|q|ⁿ< |q| / (n(1-|q|) <αn>|q| / (n(1-|q|) 1.2.1.Theorems about limits of sequences. 1. A sequence that has a limit is bounded; 2. A sequence can have only one limit; 3. Any non-decreasing (non-increasing) and not bounded from above (from below) sequence has a limit; 4. The limit of a constant is equal to this constant: ℓimn→∞ C=C 5. The limit of the sum is equal to the sum of the limits: ℓimn→∞(an+bn)= ℓimn→∞ an+ ℓimn→∞ bn; 6. A constant factor can be taken out of the limit sign: ℓim n→∞ (Сan)= Clim n→∞ an; 7. The limit of the product is equal to the product of the limits: ℓimn→∞ (an∙bn)= ℓimn→∞ an ∙ ℓimn→∞ bn; 8. The limit of the quotient is equal to the quotient of the limits if the divisor limit is non-zero: ℓimn→∞ (an/bn)= ℓimn→∞ an / ℓimn→∞ bn if ℓimn→∞bn≠0; 9. If bn ≤ an ≤ cn and both sequences (bn) and (cn) have the same limit α, then ℓimn→∞ an=α. Find the limit ℓimn→∞ ((3n-1)/(4n+5)). ℓimn→∞ ((3n-1)/(4n+5))= ℓimn→∞(n(3-1/n))/ (n(4+5/n)= (ℓimn→∞ 3-1/n )/ (ℓimn→∞ 4+5/n)= (ℓimn→∞ 3- ℓimn→∞ 1/n)/ (ℓimn→∞ 4+ 5 ℓimn→∞ 1/n)= (3-0)/(4 +5∙0)=3/4. 1.3 Arithmetic progression. An arithmetic progression is a sequence (an), each term of which, starting from the second, is equal to the previous one, added with the same number d, called the difference of the progression: an+1= an+ d, n=1, 2, 3… . Any member of the sequence can be calculated using the formula an= a1+ (n – 1)d, n≥1 1.3.1. Properties of an arithmetic progression 1. If d> 0, then the progression is increasing; if d< 0- убывающая; 2. Any member of the arithmetic progression, starting from the second, is the arithmetic mean of the previous and next member of the progression: an= (an-1 + an+1)/2, n≥2 3. The sum of the first n members of an arithmetic progression can be expressed by the formulas: Sn= ((2a1+ d(n-1))/2)∙n 4. The sum of n consecutive members of an arithmetic progression starting from member k: Sn= ((ak+ak+n-1)/2)∙n 5. An example of the sum of an arithmetic progression is the sum of a series of natural numbers up to n inclusive: It is known that for any n the sum Sn of members of some arithmetic progression is expressed by the formula Sn=4n²-3n. Find the first three terms of this progression. Sn=4n²-3n (by condition). Letn=1, then S1=4-3=1=a1 => a1=1; Letn=2, then S2=4∙2²-3∙2=10=a1+a2; a2=10-1=9; Since a2=a1+d, then d= a2-a1=9-1=8; Answer: 1; 9; 17. When dividing the ninth term of an arithmetic progression by the second term, the quotient is 5, and when dividing the thirteenth term by the sixth term, the quotient is 2 and the remainder is 5. Find the first term and the difference of the progression. a1, a2, a3…, an- arithmetic progression a13/a6=2 (remainderS) Using the formula for the nth term of the progression, we obtain the system of equations (a1+8d= S(a1+d); a1+12d = 2(a1+S∙d)+S ( 4a1=3d; a1=2d-S Whence 4(2d-S)=3d => Sd= 20 => d=4. Answer: a1=3; d=4. 1.4.Geometric progression. A geometric progression is a sequence (bn) whose first term is non-zero, and each term, starting from the second, is equal to the previous one multiplied by the same non-zero number q, called the denominator of the progression: bn+1= bnq, n= 1, 2, 3… . Any member of a geometric progression can be calculated using the formula: 1.4.1. Properties of a geometric progression. 1. The logarithms of the members of a geometric progression form an arithmetic progression. 2. b²n= bn-i bn+i, i< n 3. The product of the first n members of a geometric progression can be calculated using the formula: Pn= (b1∙bn)ⁿ َ ² 4. The product of the terms of a geometric progression, starting from the k-th member, and ending with the n-th member, can be calculated by the formula: Pk,n= (Pn)/(Pk-1); 5. The sum of the first n members of a geometric progression: Sn= b1((1-qⁿ)/(1-q)), q≠ 1 6. If |q|< 1, то bn→0 при n→+∞, и Sn→(b1)/(1-q), при n→+∞ Let a1, a2, a3, …, an, … be consecutive members of a geometric progression, Sn be the sum of its first n members. Sn= a1+a2+a3+…an-2+an-1+ an= a1an (1/an+a2/a1an+a3/a1an+…+an-2/a1an+an-1/a1an+1/a1)= a1an (1/an+ a2/a2an-1+…+ an-2/an-2a3+an-1/an-1a2+1/a1)= a1a2 (1/an+ 1/an-1+ 1/an-2+…+ 1/a3+1/a2+ 1/a1). 1.5. Fibonacci numbers. In 1202, a book appeared by the Italian mathematician Leonardo from Pisa, which contained information on mathematics and provided solutions to various problems. Among them was a simple, not devoid of practical value, problem about rabbits: “How many pairs of rabbits are born from one pair in one year?” As a result of solving this problem, we got a series of numbers 1, 2, 3, 5.8, 13, 21, 34, 55, 89, 144, etc. This series of numbers was later named after Fibonacci, as Leonardo was called. What is remarkable about the numbers obtained by Fibonacci? (In this series, each subsequent number is the sum of the two previous numbers). Mathematically, the Fibonacci series is written as follows: I1, I2,: In, where In = And n - 1 + In - 2 Such sequences, in which each term is a function of the previous ones, are called recurrent, or age sequences. The series of Fibonacci numbers is also recurrent, and the members of this series are called Fibonacci numbers. It turned out that they have a number of interesting and important properties. Four centuries after Fibonacci's discovery of a series of numbers, the German mathematician and astronomer Johannes Kepler established that the ratio of adjacent numbers in the limit tends to the golden ratio. Ph - the designation of the golden ratio on behalf of Phidias - a Greek sculptor who used the golden ratio when creating his creations. [If, when dividing a whole into two parts, the ratio of the larger part to the smaller is equal to the ratio of the whole to the larger part, then such a proportion is called "golden" and is approximately 1.618]. 1.5.1. Connection of Fibonacci numbers with other areas of knowledge The properties of a series of Fibonacci numbers are inextricably linked with the golden ratio and sometimes express the magical and even mystical essence of patterns and phenomena. The fundamental role of number in nature was determined by Pythagoras with his statement “Everything is a number”. Therefore, mathematics was one of the foundations of the religion of the followers of Pythagoras (Pythagorean Union). The Pythagoreans believed that the god Dionysus put the number at the basis of the world organization, at the basis of order; it reflected the unity of the world, its beginning, and the world was a multitude consisting of opposites. That which brings opposites to unity is harmony. Harmony is divine and lies in numerical ratios. Fibonacci numbers have many interesting properties. So, the sum of all numbers in the series from 1st to In is equal to the next one after one number (In + 2) without 2 units. The ratio of the Fibonacci numbers located through one in the limit tends to the square of the golden ratio, equal to approximately 2.618: An amazing property! It turns out that Ф + 1 = Ф2. The golden ratio is an irrational value, it reflects the irrationality in the proportions of nature. Fibonacci numbers reflect the wholeness of nature. The totality of these patterns reflect the dialectical unity of two principles: continuous and discrete. In mathematics, fundamental numbers and e are known, it is possible to add F to them. It turns out that all these universal irrational numbers, widespread in various patterns, are interconnected. e i + 1 = 0 - this formula was discovered by Euler and later de Moivre and named after the latter. Do not these formulas testify to the organic unity of the numbers e, Φ? About their fundamentality? 1.5.2. Using a series of Fibonacci numbers to describe animate and inanimate nature The world of animate and inanimate nature, it would seem that there is a huge distance between them, they are rather antipodes than relatives. But one should not forget that Live nature ultimately arose from the inanimate (if not on our planet, then in space) and, according to the laws of heredity, had to retain some features of its progenitor. The world of inanimate nature is, first of all, a world of symmetry, which gives stability and beauty to his creations. Symmetry has been preserved in wildlife. The symmetry of plants is inherited from the symmetry of crystals, the symmetry of which is inherited from the symmetry of molecules and atoms, and the symmetry of atoms - from the symmetry of elementary particles. characteristic feature structure of plants and their development is helicity. The tendrils of plants twist in a spiral, tissues grow in a spiral in tree trunks, seeds in a sunflower are arranged in a spiral. The movement of protoplasm in a cell is often spiral, the information carriers - DNA molecules - are also twisted into a spiral. The screw arrangement of atoms in some crystals (screw dislocations) has also been established. By the way, crystals with a helical structure have superstrength. Isn't that why wildlife preferred this species structural organization, inheriting it from inorganic substances? How can this regularity, the similarity of animate and inanimate nature be expressed? Scales pine cone arranged in a spiral, their number is 8 and 13 or 13 and 21. In sunflower baskets, seeds are also arranged in spirals, their number is usually 34 and 55 or 55 and 89. Take a look at the seashells. Once they served as houses for small shellfish, which they built themselves. Mollusks died long ago, and their houses will exist for millennia. The protrusions-ribs on the surface of the shell are called stiffeners by engineers - they dramatically increase the strength of the structure. These ribs are arranged in a spiral and there are 21 of them in any shell. Take any turtle - from a marsh to a giant sea turtle - and you will see that the pattern on their shell is similar: 13 fused plates are located on the oval field - 5 plates in the center and 8 at the edges, and about 21 plates on the peripheral border. Turtles have 5 toes on their paws. spinal column consists of 34 vertebrae. All of these values correspond to Fibonacci numbers. In the closest relative of the turtle, the crocodile, the body is covered with 55 horny plates. There are 55 dark spots on the body of the Caucasian viper. There are 144 vertebrae in her skeleton. Consequently, the development of a turtle, a crocodile, a viper, the formation of their bodies, was carried out according to the law of a series of Fibonacci numbers. A mosquito has 3 pairs of legs, 5 antennae on the head, and the abdomen is divided into 8 segments. The dragonfly has a massive body and a long thin tail. The body is divided into three parts: head, thorax, abdomen. The abdomen is divided into 5 segments, the tail consists of 8 parts. It is easy to see in these numbers the expansion of a series of Fibonacci numbers. The length of the tail, body and total length dragonflies are related to each other by the golden ratio: L tail = L dragonflies= F Mammals are the highest type of animals on the planet. The number of vertebrae in many domestic animals is equal to or close to 55, the number of pairs of ribs is about 13, the sternum contains 7 + 1 elements. A dog, a pig, a horse has 21 + 1 pairs of teeth, a hyena has 34, and one species of dolphins has 233. A series of Fibonacci numbers determines the general plan for the development of an organism, the evolution of species. But the development of the living is carried out not only by leaps, but also continuously. The body of any animal is in constant change, constant adaptation to its environment. Mutations of heredity violate the development plan. And it is not surprising that with the general predominant manifestation of Fibonacci numbers in the development of organisms, deviations from discrete quantities. This is not a mistake of nature, but a manifestation of the mobility of the organization of all living things, its continuous change. Fibonacci numbers reflect the basic pattern of growth of organisms, therefore, they must somehow manifest themselves in the structure of the human body. In a person: 1 - torso, head, heart, etc. 2 - arms, legs, eyes, kidneys 3 parts consist of legs, arms, fingers 5 fingers and toes 8 - composition of the hand with fingers 12 pairs of ribs (one pair is atrophied and present as a rudiment) 20 - the number of milk teeth in a child 32 is the number of teeth in an adult 34 - number of vertebrae The total number of bones in the human skeleton is close to 233. This list of human body parts can be continued. In their list, Fibonacci numbers or values \u200b\u200bclose to them are very often found. The ratio of adjacent Fibonacci numbers is approaching the golden ratio, which means that the ratio of numbers various bodies often corresponds to the golden ratio. Man, like other living creatures of nature, is subject to the universal laws of development. The roots of these laws must be sought deeply - in the structure of cells, chromosomes and genes, and far - in the emergence of life itself on Earth. 2. Own research. Task number 1. What number should replace the question mark 5; eleven; 23; ?; 95; 191? How did you find it? You need to multiply the previous number by 2 and add one. So we get: (23∙2)+1=47 => 47 is a number instead of a question mark. Task number 2. Find the sum Sn=1/(1∙2)+1/(2∙3)+1/(3∙4)+…+1/n(n+1) We write that 1/n(n+1)= 1/n - 1/(n+1). Then we rewrite the sum as a difference => Sn= (1-1/2)+(1/2-1/3)+(1/3-1/4)+…+(1/(n-1) – 1/n)+ (1/n - 1/(n+1))= 1-1/(n+1)==n/(n+1n). Answer: n/(n+1n). Task number 3. Using the definition of the limit of a sequence, prove that: ℓim n→∞an=a if an= (3n-1)/(5n+1); a= 3/5 Let us show that for any ε>0 there exists a number N(ε) such that |an-a|< ε, для |an-a|<|(3n-1)/(5n+1) - 3/5| = |(5(3n-1)-3(5n+1))/5(5n+1)|= |-8/5(5n+1)|= 8/5(5n+1) 8/5(5n+1)< ε =>5n+1> 8/5ε => n> (8/25ε)- 1/5 It follows from the last inequality that one can choose N(ε)= [(8/25ε)- 1/5] and for any n > N(ε) the inequality |an-a|< ε. Значит, по определению предела последовательности ℓimn→∞ (3n-1)/(5n+1)=3/5 Task number 4. Compute Limits of Numeric Sequences ℓimn→∞ (3-4n)²/(n-3)³-(n+3)²= ℓimn→∞ (9-24n+16n²)/(n³-9n²+27n-27)- (n³+9n²+27n+27)= ℓimn→∞(16n²-24n+9)/(-18n²-54)= ℓimn→∞ (16-24|n+9|n²)/((-18-54)/n²)= 16/-18= -8/9. Task number 5. Find ℓimn→∞ (tgx)/ x We have ℓimn→∞ (tgx)/ x= ℓimn→∞ (sinx)/ x ∙ 1/ (cosx)= ℓimn→∞ (sinx)/x ∙ ℓimn→∞ 1/(cosx)= 1∙1/1=1 Conclusion. In conclusion, I would like to say that it was very interesting for me to work on this topic. Since this topic is very interesting and informative. I got acquainted with the definition of a sequence, with its types and properties, with Fibonacci numbers. I got acquainted with the limit of the sequence, with progressions. Considered analytical tasks containing a sequence. I learned methods for solving tasks with a sequence, the connection of mathematical sequences with other areas of knowledge. List of used literature. 1. Mathematics. Big reference book for schoolchildren and university applicants./ DI. Averyanov, P.I. Altynov, I.I. Bavrin and others - 2nd ed. - Moscow: Bustard, 1999. The presentation "Numeric Sequences" presents educational material that provides a visual explanation of the teacher's explanation in the lesson on this topic. With the help of the presentation, the teacher can solve the learning problems more effectively. The presentation demonstrates theoretical material on the topic "Numeric Sequences", forms the concept of numerical sequences, their types, formulas associated with them. The presentation of educational material in the form of a presentation has many advantages that make it possible to improve the memorization of the material by students, to deepen their understanding of definitions and concepts. The animation effects used in the presentation help to keep students' attention on the subject being studied. Animation also improves the presentation of information, structures it, and contributes to a better understanding. Memorizing definitions and concepts improves their highlighting with the help of color and other techniques. The presentation begins with the definition of a numerical sequence. It is defined as a function of the form y=f(x), xϵN, otherwise called the natural argument function. The screen displays variants of the sequence designation y=f(n) or y 1 , y 2 ,…, y n , or (y n). The second slide presents options for how the numerical sequence is specified. As an example of the verbal way of setting, the sequence 2, 3, 5, ..., 29, ... is given. Variants of the analytical way of setting the sequence are also described. y n =n 3 are shown as examples. It is noted that the sequence itself is a sequence of numbers 1, 8, 27, 64, …, n 3, … Analytical representation of the sequence allows you to find any member of the sequence. For example, for n=9, 9=9 3=729. Also, with a known member of the sequence, you can determine its serial number - for y n =1331 you can determine that n 3 =1331, that is, its number n=11. Another example of the analytical specification of the sequence y n =C is presented. Obviously, in this sequence, all its terms are equal to C. Students already know examples of number sequences that were studied earlier - arithmetic and geometric progressions. To set such sequences, the recurrent method of setting was used. It is recalled that the arithmetic progression is given by the relation a 1 = a, and n+1 = a n + d, in which a and d are some numbers, and d is the difference of the progression. We also recall the recurrent definition of a geometric progression, in which b 1 =b, b n+1 =b n q, where b and q are some numbers, not zero, and q is the denominator of the progression. Slide 4 gives the definition of a sequence that is bounded from above. It is typical for such a sequence that all members of the sequence do not exceed a certain number. The next slide gives a general idea of the sequence bounded from above through the fulfillment of the inequality y n<=M, где число М, ограничивающее последовательность иначе называется верхней границей последовательности. Определение выделено цветом для запоминания понятия. Дается пример последовательности, что ограничена сверху - -1, -8, -27, -64, …, -n 3 , … Отмечается, что верхней границей данной последовательности является число М=-1, а также больше него. Similarly to the upper bound, the concept of the lower bound is considered. Before introducing the concept, consider what it means when a sequence is bounded from below. According to the definition given on slide 7, the sequence will be bounded from below if the values of the terms are not less than a certain number. The following is a general definition of a sequence that is bounded below, as a sequence for which there is a number whose value is always less than or equal to the values of the terms of the sequence. Otherwise, this number is called the lower bound of the sequence. The definition is highlighted in color and recommended for memorization. Slide 9 shows an example of a bottom bounded sequence. It is noted that the sequence 0,1,2,…, (n-1), … is bounded from below, and this bound is equal to 0 or less. Slide 10 demonstrates the definition of a bounded sequence as a numerical sequence bounded both above and below. An example is the sequence -1, -1/4, -1/9, -1/16,…, -1/n 2 ,… In this case, the upper bound of the sequence is M=0, and the lower bound is m=-1. The general term of the sequence is expressed by the formula y n =-1/n 2 . The sequence is given analytically y n =-1/х 2 , where хϵN. In the figure, a graph of such a function is constructed, demonstrating a set of points that satisfy the condition and represent a numerical sequence. Next, the geometric meaning of the notion of boundedness of a sequence is revealed. It is noted that boundedness means that all numbers of the sequence lie on a certain segment of the numerical axis. The figure shows an example of the sequence described in the previous slide. On the numerical axis, a segment containing the values of the members of the sequence is highlighted. Slide 12 defines an ascending sequence. It is noted that the sequence will be increasing if the condition y 1 The definition of a decreasing sequence is described on slide 14. It is noted that the condition for determining such a progression is y 1 >y 2 >y 3 >…>y n >y n+1 >… 5, …, 1/(2n-1), … it is obvious that the condition 1>1/3>1/5>…>1/(2n-1)>1/2(n+1)-1 >… Slide 15 also notes that decreasing and increasing sequences make up a series of monotonic sequences. The last slide gives examples of sequences whose type needs to be determined. Thus, the sequence -1,2,-3,4,…,(-1) n n, … does not increase or decrease, that is, it is not monotonic. The sequence y n =3 n is monotonically increasing. At the same time, it is noted that sequences of the form y n =a n increase for a>1. In the third example, it is noted that the sequence y n =(1/5) n is decreasing. In the general case, the sequence y n =а n is decreasing for any 0<а<1. The "Number Sequences" presentation can be used during a traditional algebra lesson to increase its effectiveness. Also, this material will help to provide clarity of explanation in the course of distance learning.