Algebraic or arithmetic progression is a kind of ordered numerical sequence whose properties are studied in a school algebra course. This article discusses in detail the question of how to find the sum of an arithmetic progression.
What is this progression?
Before proceeding to the consideration of the question (how to find the sum of the arithmetic progression), it is worth understanding what will be discussed.
Any sequence of real numbers that is obtained by adding (subtracting) a certain value from each previous number is called an algebraic (arithmetic) progression. This definition in translation into the language of mathematics takes the form:
a i = a i-1 + d.
Here i is the serial number of an element of the series a i . Thus, knowing only one initial number, you can easily restore the entire series. The parameter d in the formula is called the difference of the progression.
It can be easily shown that for the series of numbers under consideration the following equality holds:
a n = a 1 + d * (n - 1).
That is, to find the value of the nth element in order, add the difference d to the first element a 1 n-1 times.
What is the sum of the arithmetic progression: formula
Before giving a formula for the indicated sum, it is worth considering a simple special case. Given a progression of natural numbers from 1 to 10, you need to find their sum. Since there are few terms in the progression (10), it is possible to solve the problem head on, that is, sum up all the elements in order.
S 10 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55.
It is worth considering one interesting thing: since each term differs from the next one by the same value d = 1, then pairwise summation of the first with the tenth, the second with the ninth, and so on will give the same result. Really:
11 = 1 + 10 = 2 + 9 = 3 + 8 = 4 + 7 = 5 + 6.
As you can see, there are only 5 of these sums, that is, exactly two times less than the number of elements in the series. Then multiplying the number of sums (5) by the result of each sum (11), you will come to the result obtained in the first example.
If we generalize these arguments, we can write the following expression:
S n = n * (a 1 + a n ) / 2.
This expression shows that it is not necessary to sum all the elements in a row, it is enough to know the value of the first a 1 and last a n , as well as the total number of terms n.
It is believed that for the first time, Gauss came up with this equality when he was looking for a solution to the problem set by his school teacher: to sum the 100 first integers.
Sum of elements m to n: formula
The formula given in the previous paragraph gives an answer to the question of how to find the sum of the arithmetic progression (first elements), but often in problems it is necessary to add up a number of numbers in the middle of the progression. How to do it?
The answer to this question is easiest by looking at the following example: let it be necessary to find the sum of the members from the mth to the nth. To solve the problem, one should present the given segment from m to n progressions in the form of a new numerical series. In this representation, the mth term a m will be the first, and a n will be under the number n- (m-1). In this case, using the standard formula for the sum, we get the following expression:
S m n = (n - m + 1) * (a m + a n ) / 2.
An example of using formulas
Knowing how to find the sum of an arithmetic progression, it is worth considering a simple example of using the above formulas.
Below is a numerical sequence, you should find the sum of its members, starting from the 5th and ending with the 12th:
-4, -1, 2, 5, ...
The given numbers indicate that the difference d is 3. Using the expression for the nth element, we can find the values of the 5th and 12th terms of the progression. It turns out:
a 5 = a 1 + d * 4 = -4 + 3 * 4 = 8;
a 12 = a 1 + d * 11 = -4 + 3 * 11 = 29.
Knowing the values of the numbers at the ends of the algebraic progression under consideration, as well as knowing which numbers in a row they occupy, we can use the formula for the sum obtained in the previous paragraph. It will turn out:
S 5 12 = (12 - 5 + 1) * (8 + 29) / 2 = 148.
It should be noted that this value could be obtained differently: first find the sum of the first 12 elements using the standard formula, then calculate the sum of the first 4 elements using the same formula, then subtract the second from the first sum.