Before we start deciding arithmetic progression problems, consider what a numerical sequence is, since an arithmetic progression is special case numerical sequence.
A numerical sequence is a numerical set, each element of which has its own ordinal number... The elements of this set are called the members of the sequence. The ordinal number of the sequence element is indicated by the index:
The first element of the sequence;
Fifth element of the sequence;
- "nth" element of the sequence, i.e. the item "in the queue" n.
There is a relationship between the value of a sequence element and its ordinal number. Therefore, we can think of a sequence as a function whose argument is the ordinal number of an element of the sequence. In other words, we can say that a sequence is a function of a natural argument:
The sequence can be set in three ways:
1 . The sequence can be set using a table. In this case, we simply set the value of each member of the sequence.
For example, Someone decided to take up personal time management, and to begin with, calculate how much time he spends on VKontakte during the week. Writing down the time in the table, he will receive a sequence consisting of seven elements:
The first line of the table contains the number of the day of the week, the second - the time in minutes. We see that, that is, on Monday, Someone spent 125 minutes on VKontakte, that is, on Thursday - 248 minutes, and, that is, on Friday, only 15.
2 . The sequence can be specified using the nth term formula.
In this case, the dependence of the value of the sequence element on its number is expressed directly in the form of a formula.
For example, if, then
To find the value of an element of a sequence with a given number, we substitute the element number into the formula of the nth term.
We do the same if we need to find the value of a function if the value of the argument is known. We substitute the value of the argument instead into the equation of the function:
If, for example, 
, then
Once again, I note that in a sequence, unlike an arbitrary numeric function, only a natural number can be an argument.
3 ... A sequence can be specified using a formula that expresses the dependence of the value of the sequence member numbered on the value of the previous members. In this case, it is not enough for us to know only the number of the sequence member in order to find its value. We need to specify the first member or the first few members of the sequence.
For example, consider the sequence 
, 
We can find the values of the members of the sequence in sequence starting with the third:
That is, each time, to find the value of the n-th member of the sequence, we go back to the previous two. This way of sequencing is called recurrent, from the Latin word recurro- come back.
Now we can define an arithmetic progression. Arithmetic progression is a simple special case of a number sequence.
Arithmetic progression is a numerical sequence, each member of which, starting from the second, is equal to the previous one, added to the same number.
The number is called difference of arithmetic progression... The difference in arithmetic progression can be positive, negative, or zero.
If title = "(! LANG: d> 0">, то каждый член арифметической прогрессии больше предыдущего, и прогрессия является !} increasing.
For example, 2; 5; eight; eleven;...
If, then each member of the arithmetic progression is less than the previous one, and the progression is diminishing.
For example, 2; -1; -4; -7; ...
If, then all members of the progression are equal to the same number, and the progression is stationary.
For example, 2; 2; 2; 2; ...
The main property of the arithmetic progression:
Let's look at the picture.
We see that
, and at the same time
Adding these two equalities, we get:
.
Divide both sides of the equality by 2:
So, each member of the arithmetic progression, starting from the second, is equal to the arithmetic mean of two neighboring ones:
Moreover, since
, and at the same time
, then
, and therefore
Each member of the arithmetic progression starting with title = "(! LANG: k> l">, равен среднему арифметическому двух равноотстоящих.
!}
Formula of the th member.
We see that for the members of the arithmetic progression, the following relations hold:
and finally
We got the formula of the nth term.
IMPORTANT! Any member of the arithmetic progression can be expressed in terms of and. Knowing the first term and the difference of the arithmetic progression, you can find any of its terms.
The sum of n members of an arithmetic progression.
In an arbitrary arithmetic progression, the sums of the members equidistant from the extreme are equal to each other:
Consider an arithmetic progression with n terms. Let the sum of n members of this progression be.
Let us arrange the members of the progression first in ascending order of numbers, and then in descending order:
Let's add in pairs:
The sum in each parenthesis is equal, the number of pairs is n.
We get:
So, the sum of n members of an arithmetic progression can be found by the formulas:
Consider solving problems for arithmetic progression.
1 . The sequence is given by the nth term formula: . Prove that this sequence is an arithmetic progression.
Let us prove that the difference between two adjacent members of the sequence is equal to the same number.
We found that the difference between two adjacent members of the sequence does not depend on their number and is constant. Therefore, by definition, this sequence is an arithmetic progression.
2 . You are given an arithmetic progression -31; -27; ...
a) Find 31 members of the progression.
b) Determine if the number 41 is included in this progression.
a) We see that;
Let's write the formula for the nth term for our progression.
In general 
In our case 
, therefore 
Size: px
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Transcript
1 1 Calculations by formulas Answers to tasks are a word, phrase, number or sequence of words, numbers. Write down your answer without spaces, commas, or other additional characters. Select an arithmetic progression from the given sequences. In your answer, write down the difference of this arithmetic progression. (an): anan + 1 = 4 (bn): 4, 5, 7, 10 1 (cn): 1, 1 2, 1 3, 1 4, 2 formula t F = l, 8t c +32, where tc temperature in degrees Celsius, t F temperature in degrees Fahrenheit. What is the Celsius temperature of 140 Fahrenheit? Round your answer to the nearest tenth. 2 3 The boy throws pebbles into the well and calculates the distance to the water in the well using the formula s = 5t 2, where s is the distance in meters, t is the fall time in seconds. After the rain, the water level rose. Find how many meters the water level in the well has risen if the fall time was 0.6 s before the rain, and the measured time changed by 0.2 s after the rain. 3 4 The formula t F = l, 8t c + 32, where t c is the temperature in degrees Celsius, t F is the temperature in degrees Fahrenheit, allows you to convert the temperature value from Celsius to Fahrenheit. What is the Celsius temperature at 23 Fahrenheit? 4 5 DC power (in watts) is calculated by the formula P = I 2 R, where I is the current (in amperes), R is the resistance (in ohms). Using this formula, find the resistance R (in ohms) if the power is 28 W and the current is 2 A. 5 6 use the formula t F = l, 8t c + 32, where tc is the temperature in degrees Celsius, t F corresponds to 50 on the Celsius scale? 6 7 formula t F = 1.8t c +32, where t c is the temperature in degrees Celsius, t F is the temperature in degrees Fahrenheit. What is the Celsius temperature of 50 Fahrenheit? Round your answer to the nearest tenth. 7 8 In a construction company, the cost (in rubles) of laying paving slabs on the paths of a city park is calculated using the formula с = n, where n is the number of square meters of tiles. Using this formula, calculate the cost of laying on an area of 90 m 2. Indicate your answer in rubles. 8 9 DC power (in watts) is calculated by the formula P = I 2 R, where I is the current strength (in amperes), R is the resistance (in ohms). Using this formula, find the resistance R (in ohms) if the power is 211.25 W and the current is 6.5 A ID_9629 1/7 neznaika.pro 10
2 use the formula t F = 1.8t c +32, where t c is the temperature in degrees Celsius, t F corresponds to -85 on the Celsius scale? 11 Centripetal acceleration when moving around a circle (in m / s2) can be calculated by the formula a = w 2 R, where w is the angular velocity (in s -1), and R is the radius of the circle. Using this formula, find the radius R (in meters) if the angular velocity is 9.5 s -1, and the centripetal acceleration is 180.5 m / s The oscillation period of the mathematical pendulum (in seconds) can be approximately calculated by the formula T = 2 l , where I is the length of the thread in meters. Using the formula, find the length of the pendulum (in meters), the oscillation period of which is 12 seconds, use the formula t F = 1.8t c +32, where t c is the temperature in degrees Celsius, t F corresponds to -70 on the Celsius scale? The centripetal acceleration when moving in a circle (in m / s 2) can be calculated by the formula a = w 2 R, where w is the angular velocity (in s -1), and R is the radius of the circle. Using this formula, find the radius R (in meters) if the angular velocity is 8.5 s -1 and the centripetal acceleration is 289 m / s formula t F = 1.8t s +32, where tc is the temperature in degrees Celsius, t F temperature in degrees Fahrenheit. What is the Celsius temperature of 113 Fahrenheit? Round the answer to tenths The flight range of a body thrown with an initial velocity v0 and a direction of velocity α is calculated by the formula s = v 2 0 sin 2α g where g is the acceleration of gravity. Find sin 2α, if s = 25 m, v 0 = 20 m / s, g = 10 m / s the cost of a taxi ride (in rubles) lasting more than 5 minutes is calculated using the formula С = (2-5), where t is the duration of the trip, expressed in minutes. Use this formula to calculate the cost of a 10 minute ride. Indicate the answer in rubles the formula t F = 1.8t c +32, where t c is the temperature in degrees Celsius, t F is the temperature in degrees Fahrenheit. What is the Celsius temperature of -112 Fahrenheit? Round the answer to tenths use the formula t F = 1.8t c + 32, where t c is the temperature in degrees Celsius, t F corresponds to 90 on the Celsius scale? 19 ID_9629 2/7 neznaika.pro
3 20 S = d 1 d 2 sinα The area of the quadrilateral can be calculated by formula 2, where d 1 and d 2 are the lengths of the diagonals of the quadrilateral, and the angle between the diagonals. Using this formula, find the length of the diagonal d 2 if d 1 = 10, sinα = 1/11, and S = The oscillation period of the mathematical pendulum T (in seconds) can be approximately calculated by the formula T = 2 l, where l is the length of the thread (in meters). Using this formula, find the length of the pendulum thread (in meters), the oscillation period of which is 3 seconds.The maximum height h to which the body rises, thrown with the initial 2 v 0 sin 2 α h = speed v 0 and the direction of the speed α, is calculated by the formula 2g where g is the acceleration due to gravity. Find v 0 (in m / s) if h = 0.45, sin α = 0.2, g = 10 m / s From the given sequences, select an arithmetic progression. In the answer, write down the difference of the arithmetic progression. (an): anan + 1 = 3 23 (bn): 4, 6, 7, 8, (cn): 1, 1 5, 1 10, 1 15, 1 20, 24 use the formula t F = 1.8t c +32, where tc is the temperature in degrees Celsius, t F corresponds to 35 on the Celsius scale? In a construction company, the cost (in rubles) of laying paving slabs on the paths of a city park is calculated using the formula c = n, where n is the number of square meters of tiles. Using this formula, calculate the cost of laying on an area of 60 m 2. Indicate the answer in rubles using the formula t F = 1,8t c +32, where t c is the temperature in degrees Celsius, t F corresponds to 25 on the Celsius scale? The boy throws pebbles into the well and calculates the distance to the water in the well using the formula s = 5t 2, where s is the distance in meters, t is the fall time in seconds. After the rain, the water level rose. Find how many meters the water level in the well has risen, if before the rain the fall time was 0.3 s, and after the rain the measured time changed by 0.2 s DC power (in watts) is calculated by the formula P = I 2 R, where I amperage (in amperes), R resistance (in ohms). Using this formula, find the resistance R (in ohms) if the power is 15.75 W and the current is 1.5 A. the cost of a taxi ride (in rubles) for more than 5 29 ID_9629 3/7 neznaika.pro
4 minutes is calculated using the formula С = (t - 5), where t is the duration of the trip, expressed in minutes. Use this formula to calculate the cost of an 8-minute ride. Please indicate your answer in rubles. 30 The oscillation period of a mathematical pendulum (in seconds) can be approximately calculated by the formula T 2 l where I is the length of the thread in meters. Using the formula, find the oscillation period of the pendulum (in meters), the length of which is 9 m. the cost of a taxi ride (in rubles) lasting more than 5 minutes is calculated by the formula C = (t - 5), where t is the duration of the trip, expressed in minutes. Use this formula to calculate the cost of a 14-minute ride. Please indicate your answer in rubles. 31 ID_9629 4/7 neznaika.pro
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I) Physics 3. Calculations by formulas Part. FIPI. To convert the temperature value from Celsius (t C) to Fahrenheit (t F), use the formula t F =, 8t +3, where C is degrees Celsius, F is degrees Fahrenheit.
In this task, you just need to substitute the values that are given in the condition into the formula. The formula is also given in the statement. The hardest part of this task is the calculations. Most likely, you will have to multiply or divide by
TASKS 3 OGE ON MATHEMATICS CALCULATIONS BY FORMULAS) To convert the temperature value in Celsius (t, C) to Fahrenheit (t, F), use the formula t, 8t 3, where C is degrees Celsius, F is degrees Fahrenheit.
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TASK 4 Actions with formulas 1. Newton's second law can be written as F = ma, where F is the force (in newtons) acting on the body, m is its mass (in kilograms), and the acceleration with which the body moves (in m / s
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Actions with formulas 1. Find m from the equality F = ma, if F = 84 and a = 12. 2. The geometric mean of three numbers and is calculated by the formula Calculate the geometric mean of the numbers 12, 18, 27. 3. In the company
Option 29 Part 1 1 Find the value of the expression 6.8 4.7 1.4 1 2 Is the number a marked on the coordinate line? 2 Which statement about this number is correct? 1) 4 a> 0 2) 3) 4) a 6> 0 3 Value
1 Graphs of functions Answers to tasks are a word, phrase, number or sequence of words, numbers. Write down your answer without spaces, commas, or other additional characters. The pictures show
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Option 15 Part 1 1 Find the value of the expression 0, 1 + 9 Which of these numbers belongs to the interval? 1) 7) 8 3) 4 4) 61 3 Which of the expressions below for any values of n is equal to the product
Option 5 Part 1 1 Find the value of the expression 1 2 Indicate which inequality follows from the inequality a> b. 2 1) 2) 3) 4) 3 Find the value of the expression 8 6 2 2 3 1) 576 2) 24 3) 96 4) 24 3 3 4 Solve the equation
Option 4 Part 1 1 Find the meaning of the expression 8 15 + 3 10 9 1 2 Choose the correct statement about the numbers a and b located on the number line. 2 1) a b> 0 2) 3) 4) 3 Find the value of the expression
Option 3 Part 1 1 Find the values of the expressions. In the answer, write down the number of the smallest value found. 1 1) 2 It is known that m> n. Please enter an invalid inequality. 2 1) 4) 3 Select the correct statement
Option 6 Part 1 1 Find the value of the expression 1 2 Indicate the inequality that follows from the inequality m< n. 1) m + n < 0 2) n/m < 1 3) m - n < 0 4) m/n >1 2 3 Find the meaning of the expression 2 2 5 3 3 10
1 Analysis of charts, tables, graphs Answers to tasks are a word, phrase, number or sequence of words, numbers. Write down your answer without spaces, commas, or other additional characters.
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OPTION 30 Part 1 Module "Algebra" 1. Find the value of the expression 9.8 3.9 2.8 2. The numbers a, b, c are marked on the number line. a О Indicate the number of the correct statement. 1)> b ^ 3) a + c> 1 2)> b ^ 4) ^< 0 о
Option 26 Part 1 1 Find the values of the expressions. In the answer, write down the number of the smallest value found. 1 1) 4.3 4 5 2) 5 4 3 2 3 3) 1.5 0.3 + 2.2 2 The numbers p and k are marked on the coordinate line. Which
Option 9 Part 1 1 Find the value of the expression 1 2 The numbers m and n are marked on the coordinate line. Which of the following numbers is the largest? 2 1) m - n 2) -m + n 3) -2m 4) 1 / 2n + 1 3 The value of which of the expressions
Option 7 Part 1 1 Find the value of the expression (0.51) 2 + 0.0399-5 1 2 Points corresponding to numbers a and b are marked on the coordinate line. 2 Which of the following numbers is the largest? 1) -a 2) b 3)
Option 10 Part 1 1 Find the values of the expressions. In the answer, write down the number of the largest value found. 1 5.7 3 4) 1 3 1 3 6.4 0.4 +, 8 It is known that m> n. Indicate the correct inequality. m n> 1) 4)
Option 9 Part 1 1 Find the value of the expression, 4 1,3 0,7 1 2 One of the numbers 4; ; nine ; 13 is marked on a straight line. 2 What number is it? 1) 2) 4) 4 9 13 3 Find the value of the expression 3 5 3 7 1) -3 2) 3 3 1
Option 12 Part 1 1 Find the value of the expression 0.8 1 1 + 1 7 2 Points corresponding to the numbers a and b are marked on the coordinate line. 2 Which of the following numbers is the largest? 1) - a 2) b - a 3) b
Option 4 Part 1 1 Find the values of the expressions. In the answer, write down the number of the largest value found. 1 1) 2) 3) 2 Choose the correct statement about the numbers a and b on the number line.
1 Figures on a square lattice Answers to tasks are a word, phrase, number or sequence of words, numbers. Write down your answer without spaces, commas, or other additional characters. Checkered
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Option 3 Part 1 1 1 Find the value of the expression 7.6 8 (5.2) 2 Which of the following numbers is enclosed between the numbers 8 15 and 12 19 2 1) 0.6 2) 0.7 3) 0.8 4) 0 , 9 3 9 3 9 6 3 Find the value of the expression 9 7 1)
1 Squares of figures Answers to tasks are a word, phrase, number or sequence of words, numbers. Write down your answer without spaces, commas, or other additional characters. The sum of two isosceles angles
Variant Part 1 1 Find the value of the expression 9 17 11 34 17 1 The numbers x and y are marked on the coordinate line. Which of the following statements is incorrect for these numbers? 1)) x y> 0 3) x + y> 0 4) 3 Value
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Option 1 Part 1 1 Find the value of the expression 1 2 3 4 It is known that m> n. Indicate the correct inequality. 1) 2) 3) 4) Find the value of the expression 1) 2) 3) 4) Solve the equation 2 3 4 5 The figure shows
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1 Practical problems in geometry Answers to the tasks are a word, phrase, number or sequence of words, numbers. Write down your answer without spaces, commas, or other additional characters. Fire department
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