Presentation systematization arithmetic progression and geometric. Presentation - Arithmetic and Geometric Progressions

Slide 22

1. Solution: b2 = 3q, b3 = 3q2, q = -5; -4; -3; -2; -13; -15; 75 3; -12; 48; ... 3; -nine; 27; ... 3; -6; 12; ... 3; -3; 3; ... Answer:

Slide 23

2. Three numbers form an arithmetic progression. If you add 8 to the first number, you get a geometric progression with the sum of 26 members. Find these numbers. Solution: Answer: -6; 6; 18 or 10; 6; 2

Slide 24

3. The equation has roots, and the equation has roots. Determine k and m if the numbers are consecutive members of an increasing geometric progression. hint Solution: - geometric progression Answer: k = 2, m = 32

Slide 25

Vieta's theorem: the sum of the roots of the reduced quadratic equation is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term.

Slide 26

literature

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Abstract

MBOU "Voronezh Cadet

school them. A.V. Suvorov "

Semyaninova E.N.

The ability to solve problems is a practical art,

like swimming or skiing, or

imitating selected models and constantly exercising.

Find the sum of eleven members of an arithmetic progression, the first term of which is - 5, and the sixth is - 3.5.

Answer: 77dm

Answer: 18 tons

Answer: 4 seconds

Snail

meters. (Slide 12)

Answer: 30 days

Answer: 1900

Another example.

64 6 -1 6 – (-1) = 7

It's not hard to figure out:

2-3∙ 27 = 24, 26: 2-1 = 27

V. Crossnumber. (Slide 19-20)

Group work.

Horizontally:

;

127; -119; …;

Vertically:

You are given a geometric progression 3; b2; b3;…, the denominator of which is an integer. Find this progression if

12q2 + 72q +35 = 0

Hence, q = -5; -4; -3; -2; -1

Answer: -6; 6; 18 or 10; 6; 2

k and m

By Vieta's theorem

Searched numbers: 1; 2; 4; eight.

Answer: k = 2, m = 32

Vii. Homework.

Solve the problems.

Literature:

Algebra Grade 9. Tasks for training and development of students / comp. Belenkova E.Yu. "Intellect - Center". 2005.

Library of the journal "Mathematics at school". Issue 23. Mathematics in puzzles, crosswords, teawords, cryptograms. Khudadatova S.S. Moscow. 2003.

Maths. Supplement to the newspaper "First September". 2000. No. 46.

Multilevel didactic materials in algebra for grade 9 / comp. THOSE. Bondarenko. Voronezh. 2001.

MBOU "Voronezh Cadet

school them. A.V. Suvorov "

Semyaninova E.N.

Topic "Arithmetic and Geometric Progression".

1) summarize information on progressions; improve the skills of finding the nth term and the sum of the first n terms of these progressions using formulas; solving problems in which both sequences are used;

2) continue the formation of practical skills;

3) develop the cognitive interest of students, teach them to see the connection between mathematics and the surrounding life.

The ability to solve problems is a practical art,

like swimming or skiing, or

playing the piano; you can only learn this,

imitating selected models and constantly exercising.

I. Organizing time... Explanation of the objectives of the lesson. (Slide 2)

II. Warm up. In a world of interesting things. (Slide 3-6)

The French word for "dessert" means sweet food served at the end of a meal. Some desserts, cakes and ice cream are also named after French. For example, ice cream "plombir" got its name from the French city of Plombier. Where it was first made according to a special recipe.

Using the answer you found and the data in the table, find out how the French word meringue (light cake made from whipped egg whites and sugar) is translated?

Find the sum of eleven members of an arithmetic progression, the first term of which is - 5, and the sixth is - 3.5.

The French word "meringue" in translation means a kiss. The second of the suggested words - "lightning", is a translation of the French word "eclair" (custard cake with cream inside).

III. Progressions in life and everyday life. (Slide 7)

Progression problems are not abstract formulas. They are taken from our very life, are connected with it and help to solve some practical issues.

The vertical truss rods have the following length: the smallest one is 5 dm, and each next one is 2 dm longer. Find the length of seven such rods. (Slide 8)

Answer: 77dm

Under favorable conditions, the bacterium multiplies so that it divides by three in 1 second. How many bacteria will there be in a test tube in 5 seconds? (Slide 9)

The truck transports a batch of crushed stone weighing 210 tons, daily increasing the transportation rate by the same number of tons. It is known that during the first day 2 tons of crushed stone were transported. Determine how many tons of rubble were transported on the ninth day if all the work was completed in 14 days. (Slide 10)

Answer: 18 tons

The body falls from the tower, 6 m high. In the first second, it travels 2 m, for each next second - 3 m more than the previous one. How many seconds will the body take to the ground? (Slide 11)

Answer: 4 seconds

The snail crawls from one tree to another. Each day she crawls the same distance more than the previous day. It is known that during the first and last days the snail crawled a total of 10 meters. Determine how many days the snail spent all the way if the distance between the trees is 150

meters. (Slide 12)

Answer: 30 days

A truck left point A at a speed of 40 km / h. At the same time, a second car set off from point B towards him, which covered 20 km in the first hour, and each next one traveled 5 km more than the previous one. In how many hours will they meet if the distance from A to B is 125 km? (Slide 13) Answer: 2 hours

The amphitheater consists of 10 rows, with 20 more seats in each next row than in the previous one, and 280 seats in the last row. How many people can the amphitheater hold? (Slide 14)

Answer: 1900

IV. A little history. (Slide 15-16)

Problems on geometric and arithmetic progressions are found among the Babylonians, in Egyptian papyri, in the ancient Chinese treatise "Mathematics in 9 Books". Archimedes, apparently, was the first to draw attention to the connection between progressions. In 1544 the book of the German mathematician M. Stiefel "General arithmetic" was published. Stiefel made the following table (Slide 17):

In the upper line - an arithmetic progression with a difference of 1. In the lower - a geometric progression with a denominator of 2. They are arranged so that the zero of the arithmetic progression corresponds to a unit of the geometric progression. This is a very important fact.

Now imagine that we cannot multiply and divide. It is necessary to multiply, for example, by 128. In the table above it is written -3, and above 128 it is written 7. Let's add these numbers. It turned out 4. Under 4 we read 16. This is the desired work.

Another example.

Divide 64 by. We do the same:

64 6 -1 6 – (-1) = 7

The bottom line of the Stiefel table can be rewritten as follows:

2-4; 2-3; 2-2; 2-1; 20; 21; 22; 23; 24; 25; 26; 27.

It's not hard to figure out:

2-3∙ 27 = 24, 26: 2-1 = 27

We can say that if the indicators make up an arithmetic progression, then the degrees themselves make up a geometric progression. (Slide 18)

V. Crossnumber. (Slide 19-20)

Group work.

Crossnumber is one of the types of number puzzles. Translated from english word"Crossnumber" means "cross numbers". When drawing up crossnumbers, the same principle is applied as when drawing up crosswords: one sign is inscribed in each cell, "working" on the horizontal and vertical.

One number (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) is inscribed in each square of the cross number. And to avoid confusion, task numbers are indicated by letters. The numbers to be guessed are only positive integers; such numbers cannot start from zero (i.e. 42 cannot be written as 042).

Some crossnumber assignments may seem vague and have multiple (and sometimes very many) answers. But this is the style of crossnumber. If they always gave only unambiguous answers, then this would not be a game.

Horizontally:

a) the number of odd numbers of natural numbers, starting from 13, the sum of which is 3213;

c) the sum of the first five terms of a geometric progression, the fourth term of which is 3, and the seventh is ;

e) the sum of the first six positive members of the arithmetic progression

127; -119; …;

f) the third term of the geometric progression (bn), in which the first term is 5, and the denominator g is 10;

g) the sum -13 + (-9) + (-5) +… + 63, if its terms are consecutive members of an arithmetic progression.

Vertically:

A) the sum of all two-digit numbers divisible by nine;

B) doubled the twenty-first term of the arithmetic progression, in which the first term is -5, and the difference is 3;

C) the sixth term of the sequence, which is given by the formula of the nth term

D) the difference of the arithmetic progression, if.

Vi. Solving non-standard tasks. (Slide 21)

You are given a geometric progression 3; b2; b3;…, the denominator of which is an integer. Find this progression if

b2 = 3q, b3 = 3q2, then. Let's solve the inequality.

12q2 + 72q +35 = 0

Hence, q = -5; -4; -3; -2; -1

Sequences to be searched: 3; -15; 75; ...

Three numbers form an arithmetic progression. If you add 8 to the first number, you get a geometric progression with the sum of 26 members. Find these numbers. (Slide 23).

В, с - the required numbers. Let's make a table.

By condition, the sum of three numbers forming a geometric progression is 26, i.e. , in = 6

We use the property of members of a geometric progression. We get the equation:

Answer: -6; 6; 18 or 10; 6; 2

An equation has roots and an equation has roots. Define k and m if the numbers are consecutive members of an increasing geometric progression. (Slide 24-25)

Since the numbers form a geometric progression, we have:

By Vieta's theorem

We get it since the sequence is ascending.

Searched numbers: 1; 2; 4; eight.

Answer: k = 2, m = 32

Vii. Homework.

Solve the problems.

Find a geometric progression if the sum of its first three terms is 7, and their product is 8.

Divide 2912 into 6 parts so that the ratio of each part to the next is equal to

In arithmetic progression is and. How many members of this progression should be taken to make their total 104?

Literature:

Algebra Grade 9. Tasks for training and development of students / comp. Belenkova E.Yu. "Intellect - Center". 2005.

Library of the journal "Mathematics at school". Issue 23. Mathematics in puzzles, crosswords, teawords, cryptograms. Khudadatova S.S. Moscow. 2003.

Maths. Supplement to the newspaper "First September". 2000. No. 46.

Multilevel didactic materials on algebra for grade 9 / comp. THOSE. Bondarenko. Voronezh. 2001.

Definition of arithmetic and geometric progression. Formula of the n-th term of arithmetic and geometric progression.

"Everything is relative"

Find patterns

Oral work

Arithmetic progression

1) 1, 3, 5, 7, 9, …

2) 5, 8, 11, 14, …

3) -1, -2, -3, -4, …

4) -2, -4, -6, -8, …

Geometric progression

1) 1, 2, 4, 8, …

2) 5, 15, 45, 135, …

3) 1; 0,1; 0,001;0,0001;

4) 1, 2/3, 4/9, 8/27, …

d- difference

q-denominator

Definition

Arithmetic Geometric

progression

is called an n sequence,

nonzero numbers

each member of which, starting from the second,

equal to the previous term,

folded with one

and the same number.

multiplied by one

and the same number.

Definition

• Number sequence

а 1, а 2, а 3, ... а n, .. b 1, b 2, b 3, ... b n, ...

called

arithmetic geometric

if for all natural n

equality holds

a n + 1 = a n + d b n + 1 = b n * q

arithmetic progression ascending

decreasing arithmetic progression

increasing geometric progression

descending geometric progression

Formula of the n-th member of the progression

• Let a 1 and d

a 3 = a 2 + d = a 1 + d + d = a 1 + 2d

a 4 = a 3 + d = a 1 + 3d

…………………………… ..

a n = a 1 + (n-1) d

• Let b 1 and q

b 3 = b 2 * q = b 1 * q * q = b 1 * q 2

…………………………………………… .. b n = b 1 * q n-1

To ask

arithmetic geometric

progression, it is enough to indicate it

first term and first term and

difference denominator

Make a geometric progression:

• Every day everyone with the flu

can infect four others.

1; 4; 16; 64;…

• Dima ate a bun at recess. While eating in
• intestines got 30 dysentery sticks. Across
• bacteria divide every 20 minutes (they
• are doubled).

30; 60; 120; 240;…

• Every smoker smokes on average

8 cigarettes per day. After smoking one

cigarettes in the lungs are deposited 0.0002 grams

nicotine and tobacco tar. With each

the next cigarette is the amount

doubles.

0,0002; 0,0004; 0,0008;…

Work in notebooks Exercise 1.

Given: ( b n) - geometric progression

b 1 = 5 q = 3

Find: b 3 ; b 5 .

Solution: using the formula b n = b 1 q n-1

b 3 = b 1 q 2 = 5 . 3 2 =5 . 9=45

b 5 = b 1 q 4 = 5 . 3 4 =5 . 81=405

Answer: 45; 405.

Solution

Find

nineteenth member

arithmetic

progression if

a 1 = 30 and d = - 2.

Find

eighteenth member

arithmetic

progression if

a 1 = 7 and d = 4 .

Solution:

• We will use

by the formula of the nth term:

a n = a 1 +( n -1) d .

We get:

a 18 =7 +(18 -1)∙ 4=

=7+17∙4=7+68=75

Answer: a 18 =75.

• We will use

by the formula of the nth term:

a n = a 1 +( n -1) d .

We get:

a 19 =30+(19-1)∙(- 2)=

= 30+18∙(-2)=30-36=-6

Answer: a 19 = – 6.

Work in notebooks Task 2.

Given: ( b n) - geometric progression

b 4 = 40 q = 2

Find: b 1 .

Solution: using the formula b n = b 1 q n-1

b 4 = b 1 q 3 ; b 1 = b 4 : q 3 =40:2 3 =40 : 8=5

Answer: 5.

Solution

Work in notebooks Task 3.

Given: ( b n) - geometric progression

b 1 = -2, b 4 =-54.

Find: q .

Solution: using the formula b n = b 1 q n-1

b 4 = b 1 q 3 ; -54 = (- 2) q 3 ; q 3 = -54:(-2)=27;

Answer: 3.

Solution

Math should be taught at school

also with the aim of knowledge,

here purchased were

sufficient for ordinary

the needs of life.

I.L. Lobachevsky

Biology

Each simplest unicellular animal, the ciliate, the slipper reproduces by dividing into 2 parts. How many ciliates were there initially, if after six-fold division there were 320 of them.

5 ciliates

Light industry

Yeast cell growth occurs by dividing each

cells into two parts. How many cells became after their tenfold division, if initially there were

6144 cells

Physics

There is a radioactive substance with a mass of 256 g, the mass of which is halved per day. What will be the mass of the substance on the second day? On the third? Fifth?

128; 64; 16

Ecology

Hydra reproduces by budding, and with each division, 5 new individuals are obtained. How many divisions are required to obtain 625 individuals?

4 divisions

Preparing for GIA

is neither geometric nor arithmetic progression.

Indicate it.

IN 1; 4; 16;…

Preparing for GIA

The first three terms are given number sequences... It is known that

one of these sequences

is not geometric

progression. Indicate it.

B. -3; -nine; -27; ...

AT 3; 5; -7; ...

G. -3; ; -1;…

Preparing for GIA

• Sequences (a n), (b n), (c n)

are given by the formulas of the nth term.

Match each

sequence is the correct statement.

STATEMENT

• Subsequence -

arithmetic progression

2) Sequence -

geometric progression

3) The sequence is not

is neither arithmetic,

not exponentially

• Create or find tasks that allow you to use a geometric progression; fill out their solution in a notebook.

MANGUST

Mongoose is a fluffy animal native to India.

Body length ~ 50-60cm. Gives offspring 3 times a year, in the litter an average of 4 cubs.

1 pair = 2 mongoose

in a year

4 cubs

4 cubs

4 cubs

• 1st year - 2 mongoose
• 2nd year - 12 cubs

• 3rd year - 72 cubs !!!

How many mongoose cubs will appear in the 10th year?

v 10 = 20 155 392 cubs

Arithmetic and geometric progression What topic unites the concepts:

1) Difference 2) Sum n first terms 3) Denominator 4) First term

5) Arithmetic mean

6) Geometric mean?

Arithmetic

and

geometric

progression

Ustimkina L.I. Bolshebereznikovskaya secondary school

Progression Arithmetic Geometric

Ustimkina L.I. Bolshebereznikovskaya secondary school

The word progression comes from the Latin progression.

So, progressio translates as “moving forward”.

Ustimkina L.I. Bolshebereznikovskaya secondary school

The word progress is used in other fields of science, for example, in history to characterize the process of development of society as a whole and of an individual. Under certain conditions, any process can proceed both in the forward and in the opposite direction. The opposite direction is called regression, literally - “backward movement”.

Ustimkina L.I. Bolshebereznikovskaya secondary school

LEGEND ABOUT THE CREATOR OF CHESS

The first time on the control button, the second time on the sage

Ustimkina L.I. Bolshebereznikovskaya secondary school

Problem from the exam The young man gave the girl 3 flowers on the first day, and on each subsequent day he gave 2 more flowers than on the previous day. How much money did he spend on flowers in two weeks, if one flower costs 10 rubles?

224 flowers

224 * 10 = 2240 rubles.

Ustimkina L.I. Bolshebereznikovskaya secondary school

http://uztest.ru

Complete tasks A6 and A1

Ustimkina L.I. Bolshebereznikovskaya secondary school

Eye Charger

Ustimkina L.I. Bolshebereznikovskaya secondary school

21-24 points - mark "5"

17-20 points -evaluation "4"

12-16 points - grade "3"

0-11 points - score "2"

Ustimkina L.I. Bolshebereznikovskaya secondary school

Democritus

“ Good people get more from exercise than from nature ”

Ustimkina L.I. Bolshebereznikovskaya secondary school

100,000 RUB for 1 kopeck

Ustimkina L.I. Bolshebereznikovskaya secondary school

100,000 for 1 kopeck

• The rich millionaire returned from his absence unusually joyful: he had a happy meeting on the way, which promised great benefits.
• “There are such successes,” he told his family. “I met a stranger on the way, who was not visible. And at the end of the conversation he offered such a profitable business that it took my breath away.
• Let's do, - he says, - with you such an agreement. I will bring you a hundred thousand rubles every day for a month. No wonder, of course, but the fee is trifling. On the first day I have to pay by agreement - it's ridiculous to say - only one penny.
• One penny? - I ask again.
• One kopeck, - he says. - For the second hundred thousand you will pay 2 kopecks.
• Well, - I can't wait. - And then?
• And then: for the third hundred thousand 4 kopecks, for the fourth 8, for the fifth - 16. And so a whole month, every day twice as much against the previous one.

Ustimkina L.I. Bolshebereznikovskaya secondary school

Got for

Gave

Got for

Gave

21st hundred

22nd hundred

10 485 rubles 76 kopecks

20,971 rubles 52 kopecks

23rd hundred

20,971 rubles 52 kopecks

24th hundred

41,943 p. 04 kopecks

25th hundred

167,772 p. 16 kopecks

26th hundred

335,544 p. 32 kopecks

27th hundred

128 kopecks = 1 rubles 28 kopecks

671,088 p. 64 kopecks

10th hundred

28th hundred

1,342,177 RUB 28 kopecks

29th hundred

30th hundred

2 684 354 RUB 56 kopecks

5,368,709 RUB 12 kopecks.

Ustimkina L.I. Bolshebereznikovskaya secondary school

The rich man gave: S 30

Given: b 1 = 1; q = 2; n = 30.

S 30 =?

Solution

S n =

b 30 =1∙2 29 = 2 29

S 30 =2∙2 29 – 1= 2 ∙ 5 368 709 RUB 12 kopecks – 1 kopeck =

= 10,737,418 RUB 23 kopecks.

10,737,418 RUB 23 kopecks. - 3,000,000 RUB = 7 737 418 RUB 23 kopecks. - got a stranger

Answer : 10,737,418 RUB 23 kopecks.

Ustimkina L.I. Bolshebereznikovskaya secondary school