Many have heard of arithmetic progression, but not everyone is well aware of what it is. In this article, we give an appropriate definition, and also consider the question of how to find the difference in arithmetic progression, and give a number of examples.
Mathematical definition
So, if we are talking about arithmetic or algebraic progression (these concepts define the same thing), this means that there is a certain number series satisfying the following law: every two neighboring numbers in the series differ by the same value. Mathematically, it is written like this:
a n + 1 -a n = d
Here n means the element number a n in the sequence, and the number d is the difference of the progression (its name follows from the presented formula).
What does knowledge of the difference d say? About how far the adjacent numbers are from each other. However, knowledge of d is a necessary but not sufficient condition for determining (restoring) the entire progression. You need to know another number, which can be absolutely any element of the series in question, for example, a 4 , a10, but, as a rule, they use the first number, that is, a 1 .
Formulas for determining progression elements
In general, the information above is already enough to move on to solving specific problems. Nevertheless, before the arithmetic progression is given, and it will be necessary to find its difference, we give 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 = a 1 + (n - 1) * d
Indeed, each formula can be checked by simple enumeration: if we substitute n = 1, we get the first element, if we 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, using a known pair of numbers whose numbers 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 two elements with numbers n and m be given. Using the formula obtained above, we can compose a system of two equations:
a n = a 1 + (n - 1) * d;
a m = a 1 + (m - 1) * d
To find unknown quantities, we use the well-known simple technique for solving such a system: subtract the left and right sides in pairs, the equality will remain valid. We have:
a n = a 1 + (n - 1) * d;
a n - a m = (n - 1) * d - (m - 1) * d = d * (n - m)
Thus, we excluded one unknown (a 1 ). Now we can write the final expression for the definition of 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 of the elements themselves and their serial numbers. Attention should be paid to one important point: the differences are taken between the “senior” and “younger” 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 junior 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.
Further in the article we give examples of solving problems on computing d and restoring the number series of algebraic progression. Here I would like to note one important point.
In our age of development of computer technology, many students are trying to find solutions for their tasks on the Internet, so questions of this type often arise: to find the difference of arithmetic progression online. Upon a similar request, the search engine will issue a number of web pages, having passed to which, you will need to enter the data known from the condition (this 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 set for him.
It is recommended for these reasons to independently solve such problems. Moreover, they are not complicated.
Solution without using formulas
We will solve the first problem, and 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.
Famous elements stand close together in a row. How many times do you need to add the difference d to the smallest in order to get the largest of them? Three times (adding d for the first time, we get the 7th element, the second time - the eighth, finally, the third time - the ninth). What number do you need to add to three three times to get 18? This is the number five. Really:
3 + 5 + 5 + 5 = 18
Thus, the unknown difference is d = 5.
Of course, the decision could be made using the appropriate formula, but this was not done intentionally. A detailed explanation of the solution to the problem should be a clear and vivid example of what arithmetic progression is.
A task similar to the previous one
Now we will solve a similar problem, but change the input data. So, you should find the difference in arithmetic progression if a3 = 2, a9 = 19.
Of course, you can again resort to the "head on" solution method. But since given the elements of the series, which are relatively far apart, such a method will not be entirely convenient. But using the resulting formula will quickly lead us to the answer:
d = (a 9 - a 3 ) / (9 - 3) = (19 - 2) / (6) = 17/6 ≈ 2.83
Here we rounded the final number. How much this rounding led to an error can be judged by checking the result:
a 9 = a 3 + 2.83 + 2.83 + 2.83 + 2.83 + 2.83 + 2.83 = 18.98
This result differs by only 0.1% from the value given in the condition. Therefore, the rounding used to the hundredths can be considered a successful choice.
Tasks for applying the formula for an member
Consider a classic example of the problem of determining the unknown d: find the difference in 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 immediately apply the formula for the a n term. 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 it makes no sense to check the accuracy of the calculated result, as was done in the previous paragraph.
We will 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 to know about arithmetic progression
In addition to the problems of finding an unknown difference or individual elements, it is often necessary to solve the problems of the sum of the first members of the sequence. Consideration of these problems is beyond the scope of the topic of the article; nevertheless, for completeness, we give a general formula for the sum of n numbers of a series:
∑ n i = 1 (a i ) = n * (a 1 + a n ) / 2