- 1 Multiply the number in the middle by 5. And you will find the sum of five consecutive numbers! For example, 53 X 5 = 265. Here's how to multiply these numbers in the mind:
- Divide 53 into 50 and 3.
- Now multiply 50 x 5 = 250.
- Also multiply 3 x 5 = 15.
- Now add up the results: 250 + 15 = 265.
- 2 Explanation of the method:
- Assume that the smallest number is (x - 2). Then the other numbers are (x - 1), (x), (x + 1) and (x + 2).
- Summarize: (x - 2) + (x - 1) + (x) + (x + 1) + (x + 2) = 5x, where x is the number in the middle.
Method 4 Finding the Sum of Another Number of Consecutive Numbers
- 1 To find the sum of four consecutive numbers, multiply the largest number by 4 and subtract 6 from the result.
- 2 To find the sum of six consecutive numbers, multiply the largest number by 6 and subtract 15 from the result.
- 3 To find the sum of seven consecutive numbers, multiply the largest number by 7 and subtract 21 from the result.
- 4 To find the sum of eight consecutive numbers, multiply the largest number by 8 and subtract 28 from the result.
- You can add any number (even or odd) of consecutive numbers by adding the first and last numbers, dividing the result by two, and then multiplying the result by the number of consecutive numbers, i.e. n * (a + b) / 2.
- The described method works with any odd number of consecutive numbers, but instead of “5x” you should use “(the number of consecutive numbers) x”
- For example: 6 + 7 + 8, here x = 7.
- 3 * 7 = 21 and 6 + 7 + 8 = 21
Decomposition of numbers into components
In number theory, every natural number is easily represented as components. The decomposition of elements of the natural set into prime factors allows us to express numbers in the form of a product of components. Simple factors are elements of a whole series that are divided only by themselves and by one, but their product forms the desired number. For example, 50 is easily divided into indivisibles and written as 2 × 5 × 5. However, numbers can be represented not only in the form of a product, but also in the form of a sum.
The most famous example of the expression of natural numbers as a sum is perfect and sequential numbers. Perfect numbers are mathematical objects that are written as the sum of their own divisors. For example, such objects include 6 and 28:
- when decomposing 6 into divisors, we get 1, 2, and 3, which in total gives 6,
- by expanding 28 into dividers, we get 1, 2, 4, 7, 14, which when added gives 28.
As the natural series grows, perfect numbers are found less and less. The first six members of the perfect sequence look like this:
6, 28, 496, 8 128, 33 550 336, 8 589 869 056.
Obviously, there are not so many perfect numbers, and mathematicians still do not know whether their limit exists or if a perfect sequence rushes to infinity.
Sequential numbers are written as the sum of consecutive members of the natural series. The natural number is the positive integers that we use when counting objects. The successive members of the series are two adjacent elements, for example, 2 and 3, 17 and 18, 178 and 179.
We can write down quite a lot of natural numbers as the sum of consecutive elements. For example, we can write the number 57 in three ways:
- 7 + 8 + 9 + 10 + 11 + 12 = 57,
- 18 + 19 + 20 = 57,
- 28 + 29 = 57.
Similarly, it is easy to write 58, 59, 60 onwards, but 64 is not a consecutive number and cannot be represented as the sum of consecutive members of the natural number.
Our online calculator allows you to represent natural numbers as a sum of consecutive. As you can see, there are several ways to express a number as a sum, so our program calculates only one method, which decomposes the number into the sum of the largest number of terms.
Even number of terms
The easiest option to interpret. We restrict ourselves to an example of the sum of numbers from 1 to 100. We divide the entire series into pairs: the first term with the last, the second with the penultimate, etc. The sum in each pair is 101, and a couple of 100: 2 = 50 pieces. Therefore, the sum of all numbers is equal. Math tutor gives a diagram
, which is clear without words.
Math tutor struggles with an odd number of terms
And if the number of terms is odd? What to do? Two ways are possible:
1) Add one more such term to the beginning or end of the series, then find the sum obtained with an even number of terms (similar to the previous case) and subtract the used appendage from the answer
2) In fact, derive the corresponding formula for arithmetic progression. For this, no special knowledge for grade 9 is needed. Below under our row in the reverse order we place the same row of the same numbers. That is, the tutor in mathematics flips the original series:
In each column we get the same amount equal to 101 - the sum of the first and last term. There will be as many such columns as there are numbers in the original (top) row. Therefore, to find the full amount with it, it is enough to multiply the column sum by the number of columns, that is. Further, the math tutor explains that the result is 2 times the desired result (the olympiad student can easily understand this even without a tutor). Then it will become obvious that it remains to be divided by 2.
The fact that all couples have the same amount is easy. And how is it easier to explain about the number of pairs, so that there is no doubt?
I think, by analogy with a smaller sample of numbers, such as from 1 to 10, so that you can display the formula and check "on the fingers"
From the experience and practice of a tutor in mathematics - the question of the number of pairs, if the number of numbers in a row is even, strong children, as a rule, do not arise. And if it arises, then, most likely, it makes no sense to focus on solving olympiad problems. If we are talking about average abilities in grades 4–5, then reducing the number of objects in combination with a direct verification of the statement (sometimes on the fingers) really saves the tutor from immersing the student in the astral :).
yes, the astral is a very suitable term)) is very similar
Summation of consecutive numbers
There are several tricks to working with sequential elements of the natural series. The first of these tricks is the addition of five consecutive numbers in a quick way, which consists in multiplying by 5 the third member of the sequence. For example, if we want to quickly add 1 + 2 + 3 + 4 + 5, we just need to multiply 3 by 5 and get 15. Let's check and enter 15 into the online calculator form:
15 = 1 + 2 + 3 + 4 + 5.
If we take the following sum of five consecutive numbers, for example, 10 + 11 + 12 + 13 + 14, then multiplying the third term by 5, we get 12 × 5 = 60. We check the number 60 for the possibility of expansion in a sequential series:
- 60 = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11,
- 60 = 10 + 11 + 12 + 13 + 14,
- 60 = 19 + 20 + 21.
As you can see, the number 60 can be easily decomposed into the sum in three ways, among which there is ours, which is expressed as the sum of five consecutive numbers.
Decomposition of numbers into the sum of consecutive elements
To solve this problem, you only need to enter a number in the form of a calculator. Let's try to decompose large numbers into consecutive terms:
- 256 is not a consecutive number,
- 404 = 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54,
- 666 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36.
Thus, you can expand a sufficiently large number of members of the natural series, since non-consecutive numbers are quite rare.