Vida y= f(x), x ABOUT N, Where N- a bunch of natural numbers(or natural argument function), 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 specifying sequences. Sequences can be specified different 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 to explain from which elements the sequence is built.
Example 1. “All terms of the sequence are equal to 1.” This means we are talking about a stationary sequence 1, 1, 1, …, 1, ….
Example 2. “A sequence consists of all prime numbers in ascending order". Thus, the given sequence is 2, 3, 5, 7, 11, …. With this method of specifying the sequence in in this example it is difficult to answer what, say, the 1000th element of the sequence is equal to.
3. The recurrent method of specifying a sequence is to specify a rule that allows you to calculate n-th member of a sequence if its previous members are known. The name recurrent method comes from the Latin word recurrent- come back. Most often, in such cases, a formula is indicated that allows one to express 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; ….
You can see 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 in this example is especially studied in mathematics because it has a number of interesting properties and applications. It is called the Fibonacci sequence, named after the 13th century Italian mathematician. It is very easy to define the Fibonacci sequence recurrently, but very difficult analytically. n The th Fibonacci number is expressed through its serial number by the following formula.
At first glance, the formula for n the th Fibonacci number seems implausible, since the formula that specifies the sequence of natural numbers alone contains square roots, but you can check “manually” the validity of this formula for the first few n.
Properties of number sequences.
A numerical sequence is a special case of a numerical function, therefore 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 combined under the common term - monotonic sequences.
Example 1. y 1 = 1; y n= n 2 – increasing sequence.
Thus, the following theorem is true (a characteristic property of an arithmetic progression). A number sequence is arithmetic if and only if each of its members, except the first (and the last in the case of a finite sequence), is equal to the arithmetic mean of the preceding and subsequent members.
Example. At what value x numbers 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. At 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 terms are non-zero and each of whose terms, starting from the second, is obtained from the previous term 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 number sequence ( b n), defined 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 the sequence is a geometric progression, then so is 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- the th term of the geometric progression has the form
b n= b 1 qn– 1 .
You can obtain a formula for the sum of terms of a finite geometric progression.
Let a finite geometric progression be given
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 technique is used: 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 .
Thus, S n q= S n +b n q – b 1 and therefore
This is the formula with umma n terms of geometric progression for the case when q≠ 1.
At q= 1 the formula need not be derived separately; it is obvious that in this case S n= a 1 n.
The progression is called geometric because each term in it, except the first, is equal to the geometric mean of the previous and subsequent terms. Indeed, since
bn=bn- 1 q;
bn = bn+ 1 /q,
hence, b n 2=bn– 1 bn+ 1 and the following theorem is true (a characteristic property of a geometric 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.
Consistency limit.
Let there be a sequence ( c n} = {1/n}. This sequence is called harmonic, since each of its terms, starting from the second, is the harmonic mean between the previous and subsequent terms. Average geometric 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. The difference is considered
Does such a thing exist? N that's for everyone n ≥ N inequality 1 holds /N ? If we take it as N any natural number greater than 1/ε , then for everyone n ≥ N inequality 1 holds /n ≤ 1/N ε , Q.E.D.
Proving the presence of a limit for a particular sequence can sometimes be very difficult. The most frequently occurring sequences are well studied and are listed in reference books. There are important theorems that allow you 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 monotonic 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| accordingly (here c– arbitrary number).
Theorem 4. If the sequences ( a n} And ( b n) have limits equal to A And B pa n + qbn) has a limit pA+ qB.
Theorem 5. If the sequences ( a n) And ( b n)have limits equal to A And B accordingly, then the sequence ( a n b n) has a limit AB.
Theorem 6. If the sequences ( a n} And ( b n) have limits equal to A And B accordingly, and, in addition, b n ≠ 0 and B≠ 0, then the sequence ( a n / b n) has a limit A/B.
Anna Chugainova
Lesson No. 32 Date ____________
Algebra
Class: 9 "B"
Topic: “Numerical sequence and methods of setting it.”
The purpose of the lesson: Students should know what a number sequence is; methods for specifying a numerical sequence; be able to distinguish between different ways of specifying number sequences.
Didactic materials: handouts, supporting notes.
Technical training aids: presentation on the topic “Number sequences”.
During the classes.
2. Setting lesson goals.
Today in class you guys will learn:
What is a sequence?
What types of sequences are there?
How is the number sequence specified?
Learn to write a sequence using a formula and its many elements.
Learn to find members of a sequence.
3.Work on the material being studied.
3.1. Preparatory stage.
Guys, let's test your logical abilities. I name a few words, and you must continue:
-Monday Tuesday,…..
- January February March…;
– Glebova L, Ganovichev E, Dryakhlov V, Ibraeva G,…..(class list);
–10,11,12,…99;
From the children’s answers, it is concluded that the above tasks are sequences, that is, some kind of ordered series of numbers or concepts, when each number or concept is strictly in its place, and if the members are swapped, the sequence will be broken (Tuesday, Thursday, Monday is simply a list of days of the week). So, the topic of the lesson is number sequence.
3.1. Explanation of new material. (Demo material)
Analyzing the students' answers, give a definition of a number sequence and show ways to assign number sequences.
(Working with the textbook p. 66 – 67)
Definition 1. The function y = f(x), xN is called a function of a natural argument or a numerical sequence and is denoted: y = f(n) or y 1, y 2, y 3, ..., y n, ... or (y n).
IN in this case the independent variable is a natural number.
Most often we will denote sequences as follows: ( A n), (b n), (With n) etc.
Definition 2. Sequence members.
The elements that form a sequence are called sequence members.
New concepts: previous and subsequent member of a sequence,
A 1 …A P. (1st and nth term of sequence)
Methods for specifying a number sequence.
Analytical method.
Any nth element sequences can be determined using a formula. (demonstration material)
Explore examples
Example 1. Sequence of even numbers: y = 2n.
Example 2. Sequence of the square of natural numbers: y = n 2 ;
1, 4, 9, 16, 25, ..., n 2, ... .
Example 3. Stationary sequence: y = C;
C, C, C, ..., C, ... .
Special case: y = 5; 5, 5, 5, ..., 5, ... .
Example 4. Sequence y = 2 n ;
2, 2 2, 2 3, 2 4, ..., 2 n, ... .
Verbal method.
The rules for specifying a sequence are described in words, without specifying formulas, or when there is no pattern between the elements of the sequence.
Example 1: Number approximationsπ.
Example 2. Sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, .... .
Example 3. Sequence of numbers divisible by 5.
Example 2. Arbitrary set of numbers: 1, 4, 12, 25, 26, 33, 39, ... .
Example 3. Sequence of even numbers 2, 4, 6, 8, 10, 12, 14, 16, ... .
Recurrent method.
The recurrent method is to specify a rule that allows you to calculate nth term sequence, if its first few members are indicated (at least one first member) and a formula that allows one to calculate its next member from the previous members. Term recurrent comes from the Latin word recurrent , which means come back . When calculating the terms of a sequence using this rule, we seem to be going back all the time, calculating the next term based on the previous one. The peculiarity of this method is that to determine, for example, the 100th member of the sequence, you must first determine all the previous 99 members.
Example 1 . a 1 =a, a n+1 =a n +0.7. Let a 1 =5, then the sequence will look like: 5; 5.7; 6.4; 7.1; 7.8; 8.5; ... .
Example 2. b 1 = b, b n +1 = ½ b n. Let b 1 =23, then the sequence will look like: 23; 11.5; 5.75; 2.875; ... .
Example 3. Fibonacci sequence. This sequence is easily specified recursively: y 1 =1, y 2 =1,y n -2 +y n -1 if n=3, 4, 5, 6, ... . It will look like:
1, 1,2, 3, 5, 8, 13, 21, 34, 55, ... . (P th term of this sequence equal to the sum two previous members)
It is difficult to define the Fibonacci sequence analytically, but it is possible. The formula by which any element of this sequence is determined looks like this:
The Italian merchant Leonardo of Pisa (1180-1240), better known by his nickname Fibonacci, was a significant mathematician of the Middle Ages. Using this sequence, Fibonacci determined the number φ (fi); φ=1.618033989.
Graphic method
Members of the sequence can be represented by dots on coordinate plane. To do this, the number is plotted along the horizontal axis, and the value of the corresponding member of the sequence is plotted along the vertical axis.
To consolidate the methods of assignment, please give several examples of sequences that are specified either verbally, or analytically, or recurrently.
Types of number sequences
(Types of sequences are practiced using the sequences listed below).
Working with the textbook pp. 69-70
1) Increasing - if each term is less than the next one, i.e. a n a n +1.
2) Decreasing – if each term is greater than the next one, i.e. a n a n +1 .
3) Infinite.
4) Final.
5) Alternating sign.
6) Constant (stationary).
An increasing or decreasing sequence is called monotonic.
–1; 2; –3; 4; –5; …
1, 4, 9, 16 ,…
–1; 2; –3; 4; –5; 6; …
3; 3; 3; 3; …; 3; … .
3; 6; 9; 12; 15; 18;…
Working with the textbook: let's do it orally No. 150, 159 pp. 71, 72
3.2. Consolidation of new material. Problem solving.
To consolidate knowledge, examples are selected depending on the level of preparation of students.
Example 1. Create a possible formula for the nth element of the sequence (y n):
a) 1, 3, 5, 7, 9, 11, ...;
b) 4, 8, 12, 16, 20, ...;
Solution.
a) This is a sequence of odd numbers. Analytically, this sequence can be given by the formula y = 2n+1.
b) This is a numerical sequence in which the subsequent element is greater than the previous one by 4. Analytically, this sequence can be given by the formula y = 4n.
Example 2. Write down the first ten elements of the sequence given recurrently: y 1 =1, y 2 =2, y n = y n -2 +y n -1, if n = 3, 4, 5, 6, ....
Solution.
Each subsequent element of this sequence is equal to the sum of the two previous elements.
Example 3. The sequence (y n) is given recurrently: y 1 =1, y 2 =2,y n =5y n -1 - 6y n -2. Define this sequence analytically.
Solution.
Let's find the first few elements of the sequence.
y 3 =5y 2 -6y 1 =10-6=4;
y 4 =5y 3 -6y 2 =20-12=8;
y 5 =5y 4 -6y 3 =40-24=16;
y 6 =5y 5 -6y 4 =80-48=32;
y 7 =5y 6 -6y 5 =160-96=64.
We get the sequence: 1; 2; 4; 8; 16; 32; 64; ..., which can be represented as
2 0 ; 2 1 ; 2 2 ; 2 3 ; 2 4 ; 2 5 ; 2 6 ... .
n = 1; 2; 3; 4; 5; 6; 7... .
Analyzing the sequence, we obtain the following pattern: y = 2 n -1 .
Example 4. Given the sequence y n =24n+36-5n 2 .
a) How many positive members does it have?
b) Find the largest element of the sequence.
c) Is there a smallest element in this sequence?
This number sequence is a function of the form y = -5x 2 +24x+36, where x
a) Find the values of the function at which -5x 2 +24x+360. Let's solve the equation -5x 2 +24x+36=0.
D = b 2 -4ac = 1296, X 1 = 6, X 2 = -1.2.
The equation for the symmetry axis of the parabola y = -5x 2 +24x+36 can be found using the formula x=, we get: x=2.4.
The inequality -5x 2 +24x+360 holds for -1.2 There are five natural numbers in this interval (1, 2, 3, 4, 5). This means that in a given sequence there are five positive elements of the sequence.
b) The largest element of the sequence is determined by the selection method and it is equal to y 2 =64.
V) Smallest element No.
3.4.Tasks for independent work
2. Determine the arithmetic operation with which the average is obtained from two extreme numbers, and instead of the * sign, insert the missing number: ,3104.62.51043.60.94 1.7*4.43.1*37.2*0, 8
3. Students solved a task in which they needed to find missing numbers. They got different answers. Find the rules by which the guys filled in the cells. Task Answer 1Answer
Definition of a numerical sequence They say that a numerical sequence is given if, according to some law, every natural number (place number) is uniquely assigned certain number(sequence member). IN general view the indicated correspondence can be depicted as follows: y 1, y 2, y 3, y 4, y 5, ..., y n, ... ... n ... The number n is the nth term of the sequence. The entire sequence is usually denoted (y n).
Analytical method of specifying numerical sequences A sequence is specified analytically if the formula of the nth term is specified. For example, 1) y n= n 2 – analytical task of the sequence 1, 4, 9, 16, … 2) y n= С – constant (stationary) sequence 2) y n= 2 n – analytical task of the sequence 2, 4, 8, 16, ... Solve 585
Recurrent method of specifying numerical sequences The recurrent method of specifying a sequence is to indicate a rule that allows you to calculate the nth term if its previous members are known 1) an arithmetic progression is given by recurrent relations a 1 =a, a n+1 =a n + d 2 ) geometric progression – b 1 =b, b n+1 =b n * q
Fastening 591, 592 (a, b) 594, – 614 (a)
Bounded from above A sequence (y n) is said to be bounded from above if all its terms are not greater than a certain number. In other words, the sequence (y n) is upper bounded if there is 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 terms are not less than a certain number. In other words, the sequence (y n) is bounded from above if there is a number m such that for any n the inequality y n m holds. m – lower limit 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="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,">
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23
Verification work Option 1Option 2 1. The number sequence is given by the formula a) Calculate the first four terms of this sequence b) Is a number a member of the sequence? b) Is the number 12.25 a member of the sequence? 2. Create a formula for the th term of the sequence 2, 5, 10, 17, 26,…1, 2, 4, 8, 16,…
Topic: Number sequence and ways to set it
Main goals and objectives of the lesson
Educational: explain to students the meaning of the concepts sequence, nth member of the sequence; introduce methods of setting a sequence.
Developmental: development of independence, mutual assistance when working in a group, intelligence.
Educational: fostering activity and accuracy, the ability to always see the good, instilling love and interest in the subject
Expected results of mastering the topic
During the lesson, they will acquire new knowledge about number sequences and how to assign them. They will learn to find the right solution, create a solution algorithm and use it when solving problems. Through research, some of their properties will be discovered. All work is accompanied by slides.
Universal educational activities, the formation of which is aimed at the educational process: the ability to work in a group, develop logical thinking, the ability to analyze, research, draw conclusions, defend one’s point of view. Teach communication and collaboration skills. The use of these technologies contributes to the development of universal methods of activity and experience among students creative activity, competence, communication skills.
Lesson Key Ideas
New approaches to teaching and learning
- dialogue training
- learning how to learn
Assessment for learning and assessment of learning
Teaching Critical Thinking
Education of talented and gifted children
Lesson type
Studying new topic
Teaching methods
Visual (presentation), verbal (conversation, explanation, dialogue), practical.
Forms of organization educational activities studying
frontal; group; steam room; individual.
Interactive teaching methods used
Peer assessment, Self-assessment, Group work, Individual work,
Assessments for learning, ICT, Differentiated learning
Application of modules
Teaching how to learn, Teaching critical thinking, Assessments for learning, Using ICT in teaching and learning, Teaching talented and gifted children
Equipment and materials
Textbook, Interactive whiteboard, overhead projector, presentation, markers, wattmat A3, ruler, colored pencils, stickers, emoticons
Lesson steps
DURING THE CLASSES
Predicted results
Creating a collaborative environment
Organizing time
(Welcoming students, identifying absentees, checking students’ readiness for the lesson, organizing attention).
Division into groups.
introduction teachers
Parable “Everything is in your hands”
Once upon a time, in one city, there lived a great sage. The fame of his wisdom spread far around him hometown, people from afar came to him for advice. But there was a man in the city who was jealous of his glory. He once came to a meadow, caught a butterfly, planted it between his closed palms and thought: “Let me go to the sage and ask him: tell me, oh wisest one, which butterfly is in my hands - alive or dead? If he says dead, I will open my palms, the butterfly will fly away, if he says alive, I will close my palms and the butterfly will die. Then everyone will understand which of us is smarter.” That's how it all happened. An envious man came to the city and asked the sage: “Tell me, oh wisest one, which butterfly is in my hands - alive or dead?” Then the sage, who was really smart person, said: “Everything is in your hands”
Full readiness of the classroom and lesson equipment for work; quickly integrating the class into the business rhythm, organizing the attention of all students
The purpose of the lesson and the educational objectives of the lesson will be clearly and unambiguously formulated together with the students.
Main part of the lesson
Preparing students for active, conscious learning.
What events in our lives happen sequentially? Give examples of such phenomena and events.
Student answers:
days of the week,
names of months,
person's age,
bank account number,
there is a successive change of day and night,
the car speeds up sequentially, the houses on the street are numbered sequentially, etc.
Task for groups:
Work in groups, differentiated approach
Each group receives its own task. After completing it, each group reports to the class, the students of group 1 begin.
Task for groups:
Students are asked to find patterns and show them with an arrow.
Assignment for students of groups 1 and 2:
1st group:
In ascending order positive odd numbers
1/2; 1/3; 1/4; 1/5; 1/6
In descending order, proper fractions with numerator equal to 1
5; 10; 15; 20; 25;
In ascending order, positive numbers that are multiples of 5
1; 3; 5; 7; 9;
Group 2: find patterns
6; 8; 16; 18; 36;
Increase by 3
10; 19; 37; 73; 145;
Alternate magnification by 2 and magnification by 2 times
1; 4; 7; 10; 13;
Increase by 2 times and decrease by 1
Group 1 answers:
In ascending order, positive odd numbers (1; 3; 5; 7; 9;)
In descending order, proper fractions with a numerator equal to 1 (1/2; 1/3; 1/4; 1/5; 1/6)
In ascending order, positive numbers that are multiples of 5 (5; 10; 15; 20; 25;)
Answers of 2 groups:
1; 4; 7; 10; 13; (Increase by 3)
10; 19; 37; 73; 145; (Increase by 2 and decrease by 1)
6; 8; 16; 18; 36; (Alternate 2x magnification and 2x magnification)
Learning new material
- What do you understand by the word even?
- Give an example?
- Now say several even numbers in a row
- Now tell us about odd numbers?
- name consecutive non-even numbers
WELL DONE!
The numbers forming a sequence are called, respectively, the first, second, third, etc., nth terms of the sequence.
The members of the sequence are designated as follows:
a1; a2; a3; a4; аn;
Sequences can be finite or infinite, increasing or decreasing.
Working on a flipchart
xn=3n+2, then
x5=3.5+2=17;
x45=3.45+2=137.
Recurrent method
A formula that expresses any member of the sequence, starting from some, through the previous ones (one or more), is called recurrent (from the Latin word recurro - return).
For example, the sequence specified by the rule
a1=1; аn+1= аn +3
can be written with an ellipsis:
1; 4; 7; 10; 13;
Physical training 1,2,3,4,5,6,7, ...
4. Consolidation of the studied material (pair work, differentiated approach)
Each group receives individual task which they perform independently. When completing tasks, the children discuss the solution and write it down in a notebook.
Given sequences:
аn=n4 ; аn=(-1)nn2 ; аn=n +4; аn=-n-4; аn=2n -5; аn=3n -1.
Assignment for students of group 1: Sequences are given by formulas. Fill in the missing members of the sequence:
1; ___; 81; ___; 625; ...
-1; 4; ___; ___; -25;
5; ___; ___; ___; 9;
___; -6; ___; ___ ; -9;
___; ___; 3; 11; ___;
2; 8; ___; ___; ___;
Exercise:
Write down the first five terms of the sequence given by the formula of its nth term.
Assignment for group students:
Determine what numbers the members of these sequences are and fill out the table.
Positive and negative numbers
Positive numbers
Negative numbers
Working with textbooks No. 148, No. 151
Verification work
1. The sequence is given by the formula an=5n+2. What is its third term equal to?
a) 3 b) 17 c) 12 d) 22
2. Write down the first 5 terms of the sequence given by the formula an=n-3
a) -3,-2,-1,0,1 b) -2,-1,0,1,2
c) 0,-2,-4,-16,-50 d) 1,2,3,4,5
3. Find the sum of the first 6 terms of the number sequence: 2,4,6,8,
a) 66 b) 36 c) 32 d) 42
4. Which of the following sequences is infinitely decreasing:
a) b) 2,4,6,8,
c) d)
Answers: 1) b 2) b 3) d 4) d
Live communication with the teacher
Students find answers to the questions posed.
Students learn to analyze and draw conclusions.
Knowledge is formed of how to solve a system of inequalities with one variable
Correct answers in the process of dialogue, communication, student activity
Students complete the task
Solve on your own, check on slides.
They won’t be afraid of making mistakes; everything will become clear on the slides.
www. Bilimland.kz
Students confer, working in a group, consult with the teacher, gifted children
Students in pair work confer and find the correct solutions to the task.
Students evaluate the work of another group and give a grade. The results show that the material studied has been mastered.
The reproductive activity of a student is, first of all, the activity of a student that reproduces according to a certain algorithm, which leads to the required result.
Reflection
Summing up
So, we have looked at the concept of a sequence and how to define it.
Give examples of a number sequence: finite and infinite.
What methods of setting a sequence do you know?
What formula is called recurrent?
Summarize the lesson and note the most active students. Thank students for their work in class.
Students stick notes on stickers,
about what they learned
what new did they learn?
how did you understand the lesson?
did you like the lesson?
how they felt in the lesson.
Homework.
9 №150, №152
Correct answers during the dialogue, student activity
There will be no difficulties when doing homework
Atyrau region
Indersky district
Esbol village
school named after Zhambyl
mathematic teacher
highest category,
certified teacher
I advanced level
Iskakova Svetlana Slambekovna
| Lesson No. 32 ALGEBRA | Mathematics teacher, first category Olga Viktorovna Gaun. East Kazakhstan region Glubokovsky district KSU "Cheremshanskaya" high school» |
| Subject: Number sequence and methods for specifying it |
|
| Main goals and objectives of the lesson | Educational: Explain to students the meaning of the concepts “sequence”, “nth member of the sequence”; introduce methods of setting a sequence. Developmental I: development of logical thinking skills; development of computing skills; cultural development oral speech, development of communication and cooperation.Educational : education of observation, instilling love and interest in the subject. |
| Expected results of mastering the topic | During the lesson, they will acquire new knowledge about number sequences and how to assign them. They will learn to find the right solution, create a solution algorithm and use it when solving problems. Through research, some of their properties will be discovered. All work is accompanied by slides. The use of ICT will make it possible to conduct a lively lesson, complete a large amount of work, and the children will have sincere interest and emotional perception. Gifted students will give a presentation on Fibonacci numbers and the golden ratio. Universal educational activities, the formation of which is aimed at in the educational process: the ability to work in pairs, develop logical thinking, the ability to analyze, research, draw conclusions, and defend one’s point of view. Teach communication and collaboration skills. The use of these technologies contributes to the development of students’ universal methods of activity, creative experience, competence, and communication skills. |
| Lesson Key Ideas | New approaches to teaching and learning Dialogue training Learning how to learn Teaching Critical Thinking Education of talented and gifted children |
| Lesson type | Learning a new topic |
| Teaching methods | Visual (presentation), verbal (conversation, explanation, dialogue), practical. |
| Forms of organization of educational activities of students | frontal; steam room; individual. |
DURING THE CLASSES
Organizing time
(Welcoming students, identifying absentees, checking students’ readiness for the lesson, organizing attention).
Lesson motivation.
“Numbers rule the world,” said ancient Greek scientists. "Everything is a number." According to their philosophical worldview, numbers govern not only measure and weight, but also phenomena occurring in nature, and are the essence of the harmony that reigns in the world. Today in class we will continue to work with numbers.
Introduction to the topic, learning new material.
Let's test your logical abilities. I name a few words, and you must continue:
–Monday Tuesday,…..
– January February March…;
– Aliev, Gordeeva, Gribacheva... (class list);
–10,11,12,…99;
Conclusion: These are sequences, that is, some ordered series of numbers or concepts, when each number or concept stands strictly in its place. So, the topic of the lesson is consistency.
Today we willtalk about the types and components of number sequences, as well as ways to assign them.We will denote the sequences as follows: (аn), (bn), (сn), etc.
And now I offer you the first task: in front of you are some numerical sequences and a verbal description of these sequences. You need to find the pattern of each row and correlate it with the description. (show with arrow)(Mutual check)
The series we have considered are examplesnumber sequences .
The elements that form a sequence are calledmembers of the sequence
Andare called respectively first, second, third,...n- numeric members of the sequence. The members of the sequence are designated as follows:A
1
; A
2
; A
3
; A
4
; … A
n
; Where
n
- number
, under which given number is in sequence.
The following sequences are recorded on the screen:
(
Using the listed sequences, the notation form of the sequence member a is worked out
n
, and the concepts of previous and subsequent terms
)
.
3; 6; 9; 12; 15; 18;…
5, 3, 1, -1.
1, 4, 9, 16
,…
–1; 2; –3; 4; –5; 6; …
3; 3; 3; 3; …; 3; … .
Name a 1 for each sequence, and 3 etc. Could you continue each of these rows? What do you need to know for this?
Let's look at some more concepts likesubsequent and previous .
(for example, for a 5…, and for a n ?) - recording on the slidea n +1, a n -1
Types of sequences
(
Using the sequences listed above, the skill of identifying types of sequences is developed.
)
1) Increasing - if each term is less than the next one, i.e.a
n
<
a
n
+1.
2) Decreasing – if each term is greater than the next one, i.e.a
n
>
a
n
+1
.
3) Infinite
4) Final
5) Alternating
6) Constant (stationary)
Try to defineeach species and characterize each of the proposed sequences.
Oral tasks
Name in sequence 1; 1/2; 1/3; 1/4; 1/5; … 1/n; 1/(n+1) terms a 1 ; A 4 ; A 10 ; A n ;
Is the sequence of four-digit numbers finite? (Yes)
Name its first and last members. (Answer: 1000; 9999)
Is the sequence of writing the numbers 2; 4; 7; 1; -21; -15; ...? (no, because it is impossible to detect any pattern from the first six terms)
Physical pause (also related to the topic of today’s lesson: the starry sky, the planets of the solar system... what is the connection?)
Methods for specifying sequences
1) verbal – setting a sequence by description;
2) analytical - formulan
-th member;
3) graphic - using a graph;
4) recurrent - any member of the sequence, starting from a certain point, is expressed in terms of the previous ones
Today in the lesson we will look at the first two methods. So,verbal
way. Maybe one of you will try to set some sequence?
(For example:Make a sequence of odd natural numbers
. Describe this sequence: increasing, infinite)
Analytical
method: using the formula for the nth term of the sequence.
The general term formula allows you to calculate the term of a sequence with any given number. For example, if x n =3n+2, then
X 1 =3*1+2=5;
X 2 =3*2+2=8
X 5 =3 . 5+2=17;
X 45 =3 . 45+2=137, etc. So what is the advantageanalytical way beforeverbal ?
And I offer you the following task: formulas for specifying some sequences and the sequences themselves formed according to these formulas are given. These sequences are missing some terms. Your task,working in pairs , fill the gaps.
Self-test (the correct answer appears on the slide)
Performance creative project"Fibonacci numbers" (advance task )
Today we will get acquainted with the famous sequence:
1, 1, 2, 3, 5, 8, 13, 21, …, (Slide) Each number, starting from the third, is equal to the sum of the two previous ones. This series of natural numbers, which has its own historical name– the Fibonacci series has its own logic and beauty. Leonardo Fibonacci (1180-1240). Prominent Italian mathematician, author of The Book of Abacus. This book remained the main repository of information on arithmetic and algebra for several centuries. It was through the works of L. Fibonacci that the whole of Europe mastered Arabic numerals, the counting system, as well as practical geometry. They remained desktop textbooks almost until the era of Descartes (and this is already the 17th century!).
Watching a video.
You probably don’t quite understand what the connection is between the spiral and the Fibonacci series. So I'll show you how it turns out .
If we build two squares side by side with side 1, then on the larger side equal to 2 the other, then on the larger side equal to 3 another square ad infinitum... Then in each square, starting with the smaller one, we build a quarter of an arc, we will get the spiral we are talking about speech in the film.
In fact practical use knowledge gained in this lesson in real life big enough. Before you are several tasks from different scientific fields.
Task 1.
16, 15, 18, … (17, 20, 19)
1, 2, 2, 4, 8, … (32, 256, 8192)
33, 31, 32, … (30, 31, 29)
Task 2.
(Students' answers are written on the board: 500, 530, 560, 590, 620).
Task 3.
Task 4. Every day, each person with the flu can infect 4 people around them. In how many days will all the students in our school (300 people) get sick? (After 4 days).
Problem 5
. How many chicken cholera bacteria will appear in 10 hours if one bacterium divides in half every hour?
Problem 6
. The course of air baths begins with 15 minutes on the first day and increases the time of this procedure on each subsequent day by 10 minutes. How many days should you take air baths in the indicated mode to achieve their maximum duration of 1 hour 45 minutes? (
10)
Problem 7 . In free fall, a body travels 4.8 m in the first second, and 9.8 m more in each subsequent second. Find the depth of the shaft if a freely falling body reaches its bottom 5 s after the start of the fall.
Problem 8 . Citizen K. left a will. He spent $1,000 in the first month, and each subsequent month he spent $500 more. How much money was bequeathed to citizen K. if it is enough for 1 year of comfortable life? (45000)
Studying the following topics in this chapter of “Progression” will allow us to solve such problems quickly and without errors.
Homework: p.66 No. 151, 156, 157
Creative task: message about Pascal's triangle
Summing up. Reflection. (assessment of “increment” of knowledge and achievement of goals)
What was the purpose of today's lesson?
Has the goal been achieved?
Continue the statement
I did not know….
Now I know…
Problems on the practical application of properties of sequences (progressions)
Task 1. Continue the sequence of numbers:
16, 15, 18, …
1, 2, 2, 4, 8, …
33, 31, 32, …
Task 2. There are 500 tons of coal in the warehouse, 30 tons are delivered every day. How much coal will be in the warehouse on 1 day? Day 2? Day 3? Day 4? Day 5?
Task 3. A car, moving at a speed of 1 m/s, changed its speed by 0.6 m/s for each subsequent second. What speed will it have after 10 seconds?
Problem 4 . Every day, each person with the flu can infect 4 people around them. In how many days will all the students in our school (300 people) get sick?
Task 5. How many chicken cholera bacteria will appear in 10 hours if one bacterium divides in half every hour?
Task 6. The course of air baths begins with 15 minutes on the first day and increases the time of this procedure on each subsequent day by 10 minutes. How many days should you take air baths in the indicated mode to achieve their maximum duration of 1 hour 45 minutes?
Task 7. In free fall, a body travels 4.8 m in the first second, and 9.8 m more in each subsequent second. Find the depth of the shaft if a freely falling body reaches its bottom 5 s after the start of the fall.
Task 8. Citizen K. left a will. He spent $1,000 in the first month, and each subsequent month he spent $500 more. How much money was bequeathed to citizen K. if it is enough for 1 year of comfortable life?