Many have heard of an arithmetic progression, but not everyone is well aware of what it is. In this article, we will give the corresponding definition, and also consider the question of how to find the difference of an arithmetic progression, and give a number of examples.

Mathematical definition

So, if we are talking about an arithmetic or algebraic progression (these concepts define the same thing), then this means that there is some number series that satisfies the following law: every two adjacent numbers in the series differ by the same value. Mathematically, this is written like this:

Here n means the number of the element a n in the sequence, and the number d is the difference of the progression (its name follows from the presented formula).

What does knowing the difference d mean? About how far apart adjacent numbers are. However, knowledge of d is a necessary but not sufficient condition for determining (restoring) the entire progression. You need to know one more number, which can be absolutely any element of the series under consideration, for example, a 4, a10, but, as a rule, the first number is used, that is, a 1.

Formulas for determining the elements of the progression

In general, the information above is already enough to move on to solving specific problems. Nevertheless, before an arithmetic progression is given, and it will be necessary to find its difference, we present a couple of useful formulas, thereby facilitating the subsequent process of solving problems.

It is easy to show that any element of the sequence with number n can be found as follows:

a n \u003d a 1 + (n - 1) * d

Indeed, everyone can check this formula with a simple enumeration: if you substitute n = 1, then you get the first element, if you substitute n = 2, then the expression gives the sum of the first number and the difference, and so on.

The conditions of many problems are compiled in such a way that for a known pair of numbers, the numbers of which are also given in the sequence, it is necessary to restore the entire number series (find the difference and the first element). Now we will solve this problem in a general way.

So, let's say we are given two elements with numbers n and m. Using the formula obtained above, we can compose a system of two equations:

a n \u003d a 1 + (n - 1) * d;

a m = a 1 + (m - 1) * d

To find unknown quantities, we use a well-known simple method for solving such a system: we subtract the left and right parts in pairs, while the equality remains valid. We have:

a n \u003d a 1 + (n - 1) * d;

a n - a m = (n - 1) * d - (m - 1) * d = d * (n - m)

Thus, we have eliminated one unknown (a 1). Now we can write the final expression for determining d:

d = (a n - a m) / (n - m), where n > m

We have obtained a very simple formula: in order to calculate the difference d in accordance with the conditions of the problem, it is only necessary to take the ratio of the differences between the elements themselves and their serial numbers. Attention should be paid to one important point: the differences are taken between the "senior" and "junior" members, that is, n> m ("senior" - meaning standing further from the beginning of the sequence, its absolute value can be either more or less more "younger" element).

The expression for the difference d of the progression should be substituted into any of the equations at the beginning of the solution of the problem in order to obtain the value of the first term.

In our age of computer technology development, many schoolchildren try to find solutions for their tasks on the Internet, so questions of this type often arise: find the difference of an arithmetic progression online. Upon such a request, the search engine will display a number of web pages, by going to which, you will need to enter the data known from the condition (it can be either two members of the progression or the sum of some of them) and instantly get an answer. Nevertheless, such an approach to solving the problem is unproductive in terms of the development of the student and understanding the essence of the task assigned to him.

Solution without using formulas

Let's solve the first problem, while we will not use any of the above formulas. Let the elements of the series be given: a6 = 3, a9 = 18. Find the difference of the arithmetic progression.

Known elements are close to each other in a row. How many times must the difference d be added to the smallest one to get the largest one? Three times (the first time adding d, we get the 7th element, the second time - the eighth, finally, the third time - the ninth). What number must be added to three three times to get 18? This is the number five. Really:

Thus, the unknown difference is d = 5.

Of course, the solution could be done using the appropriate formula, but this was not done intentionally. A detailed explanation of the solution to the problem should become a clear and vivid example of what an arithmetic progression is.

A task similar to the previous one

Now let's solve a similar problem, but change the input data. So, you should find if a3 = 2, a9 = 19.

Of course, you can resort again to the method of solving "on the forehead". But since the elements of the series are given, which are relatively far apart, such a method becomes not very convenient. But using the resulting formula will quickly lead us to the answer:

d \u003d (a 9 - a 3) / (9 - 3) \u003d (19 - 2) / (6) \u003d 17 / 6 ≈ 2.83

Here we have rounded the final number. How much this rounding led to an error can be judged by checking the result:

a 9 \u003d a 3 + 2.83 + 2.83 + 2.83 + 2.83 + 2.83 + 2.83 \u003d 18.98

This result differs by only 0.1% from the value given in the condition. Therefore, rounding to hundredths used can be considered a good choice.

Tasks for applying the formula for an member

Let's consider a classic example of the problem of determining the unknown d: find the difference of the arithmetic progression if a1 = 12, a5 = 40.

When two numbers of an unknown algebraic sequence are given, and one of them is the element a 1 , then you do not need to think long, but you should immediately apply the formula for the a n member. In this case we have:

a 5 = a 1 + d * (5 - 1) => d = (a 5 - a 1) / 4 = (40 - 12) / 4 = 7

We got the exact number when dividing, so there is no point in checking the accuracy of the calculated result, as was done in the previous paragraph.

Let's solve another similar problem: we should find the difference of the arithmetic progression if a1 = 16, a8 = 37.

We use a similar approach to the previous one and get:

a 8 = a 1 + d * (8 - 1) => d = (a 8 - a 1) / 7 = (37 - 16) / 7 = 3

What else you should know about arithmetic progression

In addition to problems of finding an unknown difference or individual elements, it is often necessary to solve problems of the sum of the first terms of a sequence. The consideration of these problems is beyond the scope of the topic of the article, however, for completeness of information, we present a general formula for the sum of n numbers of the series:

∑ n i = 1 (a i) = n * (a 1 + a n) / 2

First level

Arithmetic progression. Detailed theory with examples (2019)

Numeric sequence

So let's sit down and start writing some numbers. For example:
You can write any numbers, and there can be as many as you like (in our case, them). No matter how many numbers we write, we can always say which of them is the first, which is the second, and so on to the last, that is, we can number them. This is an example of a number sequence:

Numeric sequence
For example, for our sequence:

The assigned number is specific to only one sequence number. In other words, there are no three second numbers in the sequence. The second number (like the -th number) is always the same.
The number with the number is called the -th member of the sequence.

We usually call the whole sequence some letter (for example,), and each member of this sequence - the same letter with an index equal to the number of this member: .

In our case:

Let's say we have a numerical sequence in which the difference between adjacent numbers is the same and equal.
For example:

etc.
Such a numerical sequence is called an arithmetic progression.
The term "progression" was introduced by the Roman author Boethius as early as the 6th century and was understood in a broader sense as an endless numerical sequence. The name "arithmetic" was transferred from the theory of continuous proportions, which the ancient Greeks were engaged in.

This is a numerical sequence, each member of which is equal to the previous one, added with the same number. This number is called the difference of an arithmetic progression and is denoted.

Try to determine which number sequences are an arithmetic progression and which are not:

a)
b)
c)
d)

Got it? Compare our answers:
Is an arithmetic progression - b, c.
Is not arithmetic progression - a, d.

Let's return to the given progression () and try to find the value of its th member. Exist two way to find it.

1. Method

We can add to the previous value of the progression number until we reach the th term of the progression. It’s good that we don’t have much to summarize - only three values:

So, the -th member of the described arithmetic progression is equal to.

2. Method

What if we needed to find the value of the th term of the progression? The summation would have taken us more than one hour, and it is not a fact that we would not have made mistakes when adding the numbers.
Of course, mathematicians have come up with a way in which you do not need to add the difference of an arithmetic progression to the previous value. Look closely at the drawn picture ... Surely you have already noticed a certain pattern, namely:

For example, let's see what makes up the value of the -th member of this arithmetic progression:


In other words:

Try to independently find in this way the value of a member of this arithmetic progression.

Calculated? Compare your entries with the answer:

Pay attention that you got exactly the same number as in the previous method, when we successively added the members of an arithmetic progression to the previous value.
Let's try to "depersonalize" this formula - we bring it into a general form and get:

Arithmetic progression equation.

Arithmetic progressions are either increasing or decreasing.

Increasing- progressions in which each subsequent value of the terms is greater than the previous one.
For example:

Descending- progressions in which each subsequent value of the terms is less than the previous one.
For example:

The derived formula is used in the calculation of terms in both increasing and decreasing terms of an arithmetic progression.
Let's check it out in practice.
We are given an arithmetic progression consisting of the following numbers:

Since then:

Thus, we were convinced that the formula works both in decreasing and in increasing arithmetic progression.
Try to find the -th and -th members of this arithmetic progression on your own.

Let's compare the results:

Arithmetic progression property

Let's complicate the task - we derive the property of an arithmetic progression.
Suppose we are given the following condition:
- arithmetic progression, find the value.
It's easy, you say, and start counting according to the formula you already know:

Let, a, then:

Absolutely right. It turns out that we first find, then add it to the first number and get what we are looking for. If the progression is represented by small values, then there is nothing complicated about it, but what if we are given numbers in the condition? Agree, there is a possibility of making mistakes in the calculations.
Now think, is it possible to solve this problem in one step using any formula? Of course, yes, and we will try to bring it out now.

Let's denote the desired term of the arithmetic progression as, we know the formula for finding it - this is the same formula that we derived at the beginning:
, then:

• the previous member of the progression is:
• the next term of the progression is:

Let's sum the previous and next members of the progression:

It turns out that the sum of the previous and subsequent members of the progression is twice the value of the member of the progression located between them. In other words, to find the value of a progression member with known previous and successive values, it is necessary to add them and divide by.

That's right, we got the same number. Let's fix the material. Calculate the value for the progression yourself, because it is not difficult at all.

Well done! You know almost everything about progression! It remains to find out only one formula, which, according to legend, one of the greatest mathematicians of all time, the "king of mathematicians" - Karl Gauss, easily deduced for himself ...

When Carl Gauss was 9 years old, the teacher, busy checking the work of students from other classes, asked the following task at the lesson: "Calculate the sum of all natural numbers from up to (according to other sources up to) inclusive." What was the surprise of the teacher when one of his students (it was Karl Gauss) after a minute gave the correct answer to the task, while most of the classmates of the daredevil after long calculations received the wrong result ...

Young Carl Gauss noticed a pattern that you can easily notice.
Let's say we have an arithmetic progression consisting of -ti members: We need to find the sum of the given members of the arithmetic progression. Of course, we can manually sum all the values, but what if we need to find the sum of its terms in the task, as Gauss was looking for?

Let's depict the progression given to us. Look closely at the highlighted numbers and try to perform various mathematical operations with them.


Tried? What did you notice? Correctly! Their sums are equal

Now answer, how many such pairs will there be in the progression given to us? Of course, exactly half of all numbers, that is.
Based on the fact that the sum of two terms of an arithmetic progression is equal, and similar equal pairs, we get that the total sum is equal to:
.
Thus, the formula for the sum of the first terms of any arithmetic progression will be:

In some problems, we do not know the th term, but we know the progression difference. Try to substitute in the sum formula, the formula of the th member.
What did you get?

Well done! Now let's return to the problem that was given to Carl Gauss: calculate for yourself what the sum of numbers starting from the -th is, and the sum of the numbers starting from the -th.

How much did you get?
Gauss turned out that the sum of the terms is equal, and the sum of the terms. Is that how you decided?

In fact, the formula for the sum of members of an arithmetic progression was proved by the ancient Greek scientist Diophantus back in the 3rd century, and throughout this time, witty people used the properties of an arithmetic progression with might and main.
For example, imagine Ancient Egypt and the largest construction site of that time - the construction of a pyramid ... The figure shows one side of it.

Where is the progression here you say? Look carefully and find a pattern in the number of sand blocks in each row of the pyramid wall.


Why not an arithmetic progression? Count how many blocks are needed to build one wall if block bricks are placed in the base. I hope you will not count by moving your finger across the monitor, do you remember the last formula and everything we said about arithmetic progression?

In this case, the progression looks like this:
Arithmetic progression difference.
The number of members of an arithmetic progression.
Let's substitute our data into the last formulas (we count the number of blocks in 2 ways).

Method 1.

Method 2.

And now you can also calculate on the monitor: compare the obtained values ​​​​with the number of blocks that are in our pyramid. Did it agree? Well done, you have mastered the sum of the th terms of an arithmetic progression.
Of course, you can’t build a pyramid from the blocks at the base, but from? Try to calculate how many sand bricks are needed to build a wall with this condition.
Did you manage?
The correct answer is blocks:

Workout

Tasks:

1. Masha is getting in shape for the summer. Every day she increases the number of squats by. How many times will Masha squat in weeks if she did squats at the first workout.
2. What is the sum of all odd numbers contained in.
3. When storing logs, lumberjacks stack them in such a way that each top layer contains one less log than the previous one. How many logs are in one masonry, if the base of the masonry is logs.

Answers:

1. Let us define the parameters of the arithmetic progression. In this case
   (weeks = days).

   Answer: In two weeks, Masha should squat once a day.

2. First odd number, last number.
   Arithmetic progression difference.
   The number of odd numbers in - half, however, check this fact using the formula for finding the -th member of an arithmetic progression:

   The numbers do contain odd numbers.
   We substitute the available data into the formula:

   Answer: The sum of all odd numbers contained in is equal to.

3. Recall the problem about the pyramids. For our case, a , since each top layer is reduced by one log, there are only a bunch of layers, that is.
   Substitute the data in the formula:

   Answer: There are logs in the masonry.

Summing up

1. - a numerical sequence in which the difference between adjacent numbers is the same and equal. It is increasing and decreasing.
2. Finding formula th member of an arithmetic progression is written by the formula - , where is the number of numbers in the progression.
3. Property of members of an arithmetic progression- - where - the number of numbers in the progression.
4. The sum of the members of an arithmetic progression can be found in two ways:
   
   , where is the number of values.

ARITHMETIC PROGRESSION. MIDDLE LEVEL

Numeric sequence

Let's sit down and start writing some numbers. For example:

You can write any numbers, and there can be as many as you like. But you can always tell which of them is the first, which is the second, and so on, that is, we can number them. This is an example of a number sequence.

Numeric sequence is a set of numbers, each of which can be assigned a unique number.

In other words, each number can be associated with a certain natural number, and only one. And we will not assign this number to any other number from this set.

The number with the number is called the -th member of the sequence.

We usually call the whole sequence some letter (for example,), and each member of this sequence - the same letter with an index equal to the number of this member: .

It is very convenient if the -th member of the sequence can be given by some formula. For example, the formula

sets the sequence:

And the formula is the following sequence:

For example, an arithmetic progression is a sequence (the first term here is equal, and the difference). Or (, difference).

nth term formula

We call recurrent a formula in which, in order to find out the -th term, you need to know the previous or several previous ones:

To find, for example, the th term of the progression using such a formula, we have to calculate the previous nine. For example, let. Then:

Well, now it's clear what the formula is?

In each line, we add to, multiplied by some number. For what? Very simple: this is the number of the current member minus:

Much more comfortable now, right? We check:

Decide for yourself:

In an arithmetic progression, find the formula for the nth term and find the hundredth term.

Decision:

The first term is equal. And what is the difference? And here's what:

(after all, it is called the difference because it is equal to the difference of successive members of the progression).

So the formula is:

Then the hundredth term is:

What is the sum of all natural numbers from to?

According to legend, the great mathematician Carl Gauss, being a 9-year-old boy, calculated this amount in a few minutes. He noticed that the sum of the first and last number is equal, the sum of the second and penultimate is the same, the sum of the third and the 3rd from the end is the same, and so on. How many such pairs are there? That's right, exactly half the number of all numbers, that is. So,

The general formula for the sum of the first terms of any arithmetic progression will be:

Example:
Find the sum of all two-digit multiples.

Decision:

The first such number is this. Each next is obtained by adding a number to the previous one. Thus, the numbers of interest to us form an arithmetic progression with the first term and the difference.

The formula for the th term for this progression is:

How many terms are in the progression if they must all be two digits?

Very easy: .

The last term of the progression will be equal. Then the sum:

Answer: .

Now decide for yourself:

1. Every day the athlete runs 1m more than the previous day. How many kilometers will he run in weeks if he ran km m on the first day?
2. A cyclist rides more miles each day than the previous one. On the first day he traveled km. How many days does he have to drive to cover a kilometer? How many kilometers will he travel on the last day of the journey?
3. The price of a refrigerator in the store is reduced by the same amount every year. Determine how much the price of a refrigerator decreased every year if, put up for sale for rubles, six years later it was sold for rubles.

Answers:

1. The most important thing here is to recognize the arithmetic progression and determine its parameters. In this case, (weeks = days). You need to determine the sum of the first terms of this progression:
   .
   Answer:
2. Here it is given:, it is necessary to find.
   Obviously, you need to use the same sum formula as in the previous problem:
   .
   Substitute the values:

   The root obviously doesn't fit, so the answer.
   Let's calculate the distance traveled over the last day using the formula of the -th member:
   (km).
   Answer:

3. Given: . To find: .
   It doesn't get easier:
   (rub).
   Answer:

ARITHMETIC PROGRESSION. BRIEFLY ABOUT THE MAIN

This is a numerical sequence in which the difference between adjacent numbers is the same and equal.

Arithmetic progression is increasing () and decreasing ().

For example:

The formula for finding the n-th member of an arithmetic progression

is written as a formula, where is the number of numbers in the progression.

Property of members of an arithmetic progression

It makes it easy to find a member of the progression if its neighboring members are known - where is the number of numbers in the progression.

The sum of the members of an arithmetic progression

There are two ways to find the sum:

Where is the number of values.

Where is the number of values.

The topic "arithmetic progression" is studied in the general course of algebra in schools in the 9th grade. This topic is important for further in-depth study of the mathematics of number series. In this article, we will get acquainted with the arithmetic progression, its difference, as well as with typical tasks that schoolchildren may face.

The concept of algebraic progression

A numerical progression is a sequence of numbers in which each subsequent element can be obtained from the previous one if some mathematical law is applied. There are two simple types of progression: geometric and arithmetic, which is also called algebraic. Let's dwell on it in more detail.

Imagine some rational number, denote it by the symbol a 1 , where the index indicates its ordinal number in the series under consideration. Let's add some other number to a 1, let's denote it d. Then the second element of the series can be reflected as follows: a 2 = a 1 + d. Now add d again, we get: a 3 = a 2 + d. Continuing this mathematical operation, you can get a whole series of numbers, which will be called an arithmetic progression.

As can be understood from the above, in order to find the n-th element of this sequence, you must use the formula: a n = a 1 + (n-1) * d. Indeed, substituting n=1 into the expression, we get a 1 = a 1, if n = 2, then the formula implies: a 2 = a 1 + 1*d, and so on.

For example, if the difference of the arithmetic progression is 5, and a 1 \u003d 1, then this means that the number series of the type in question looks like: 1, 6, 11, 16, 21, ... As you can see, each of its members is 5 more than the previous one .

Arithmetic progression difference formulas

From the above definition of the series of numbers under consideration, it follows that to determine it, you need to know two numbers: a 1 and d. The latter is called the difference of this progression. It uniquely determines the behavior of the entire series. Indeed, if d is positive, then the number series will constantly increase, on the contrary, in the case of negative d, the numbers in the series will increase only modulo, while their absolute value will decrease with increasing number n.

What is the difference between an arithmetic progression? Consider the two main formulas that are used to calculate this value:

1. d = a n+1 -a n , this formula follows directly from the definition of the considered series of numbers.
2. d \u003d (-a 1 + a n) / (n-1), this expression is obtained by expressing d from the formula given in the previous paragraph of the article. Note that this expression becomes indeterminate (0/0) if n=1. This is due to the fact that it is necessary to know at least 2 elements of the series in order to determine its difference.

These two basic formulas are used to solve any problem of finding the progression difference. However, there is another formula that you also need to know about.

Sum of first elements

The formula, which can be used to determine the sum of any number of members of an algebraic progression, according to historical evidence, was first obtained by the "prince" of mathematics of the XVIII century, Carl Gauss. A German scientist, while still a boy in the elementary grades of a village school, noticed that in order to add natural numbers in the series from 1 to 100, you must first sum the first element and the last (the resulting value will be equal to the sum of the penultimate and second, penultimate and third elements , and so on), and then this number should be multiplied by the number of these sums, that is, by 50.

The formula that reflects the stated result on a particular example can be generalized to an arbitrary case. It will look like: S n = n/2*(a n + a 1). Note that in order to find the specified value, knowledge of the difference d is not required if two members of the progression (a n and a 1) are known.

Example #1. Determine the difference, knowing the two terms of the series a1 and an

We will show how to apply the formulas indicated above in the article. Let's give a simple example: the difference of the arithmetic progression is unknown, it is necessary to determine what it will be equal to if a 13 \u003d -5.6 and a 1 \u003d -12.1.

Since we know the values ​​of two elements of the numerical sequence, and one of them is the first number, we can use formula No. 2 to determine the difference d. We have: d \u003d (-1 * (-12.1) + (-5.6)) / 12 \u003d 0.54167. In the expression, we used the value n=13, since the member with this ordinal number is known.

The resulting difference indicates that the progression is increasing, despite the fact that the elements given in the condition of the problem have a negative value. It can be seen that a 13 >a 1 , although |a 13 |<|a 1 |.

Example #2. Positive progression terms in example #1

Let's use the result obtained in the previous example to solve a new problem. It is formulated as follows: from what ordinal number do the elements of the progression in example No. 1 begin to take positive values?

As was shown, the progression in which a 1 = -12.1 and d = 0.54167 is increasing, so from a certain number the numbers will only take positive values. To determine this number n, it is necessary to solve a simple inequality, which is mathematically written as follows: a n>0 or, using the appropriate formula, we rewrite the inequality: a 1 + (n-1)*d>0. It is necessary to find the unknown n, let's express it: n>-1*a 1 /d + 1. Now it remains to substitute the known values ​​of the difference and the first member of the sequence. We get: n>-1*(-12.1) /0.54167 + 1= 23.338 or n>23.338. Since n can only take integer values, it follows from the obtained inequality that any terms of the series that have a number greater than 23 will be positive.

Let's check our answer by using the above formula to calculate the 23rd and 24th elements of this arithmetic progression. We have: a 23 \u003d -12.1 + 22 * ​​0.54167 \u003d -0.18326 (negative number); a 24 \u003d -12.1 + 23 * 0.54167 \u003d 0.3584 (positive value). Thus, the result obtained is correct: starting from n=24, all members of the number series will be greater than zero.

Example #3. How many logs will fit?

Here is one interesting problem: during the logging, it was decided to stack the sawn logs on top of each other as shown in the figure below. How many logs can be stacked in this way, knowing that 10 rows will fit in total?

In this way of folding logs, one interesting thing can be noticed: each subsequent row will contain one less log than the previous one, that is, there is an algebraic progression, the difference of which is d=1. Assuming that the number of logs in each row is a member of this progression, and also taking into account that a 1 = 1 (only one log will fit at the very top), we find the number a 10 . We have: a 10 \u003d 1 + 1 * (10-1) \u003d 10. That is, in the 10th row, which lies on the ground, there will be 10 logs.

The total amount of this "pyramidal" construction can be obtained by using the Gauss formula. We get: S 10 \u003d 10/2 * (10 + 1) \u003d 55 logs.