Question

В чем закономерность ? есть три числа : 1,43 4,50 и 7,40 их сумма 13,33 . Среднее арифметическое 4,4433333333333333 - так вот : 1/( 4,44333333... - 1,43) = 0.331858... ту же процедуру проделываем с двумя другими числами. далее у нас получились следующие числа : 0.331858.... 17,647058.... 0,338218.... ( если число со знаком минус переправляем на плюс ). затем следующая процедура : 1/0,331858.... + 1/17,647058..... + 1/0,338218..... = 6,02667036.... после : 6,02667036.... * 0,331858..... = 2,00 также поступаем с остальными двумя числами . В итоге выходит ( округлив ) : 2,00 106,35 2,04. Число 2,00 появляется всегда при такой операции и при всех любых других трех числах кроме нуля . Обьясните пожалуйста детально почему так происходит ? И о роли средней арифметической в этом процессе ?

Answer 1

Попробуем представить этот процесс в общем виде.
Есть 3 числа: a, b, c. Их сумма s = a+b+c, среднее d = s/3 = (a+b+c)/3
Пусть a < d, b > d, c < d. Хотя бы одно из трех чисел будет больше d.
1 шаг:
a1 = 1/(d - a) = 1/((a+b+c)/3 - a) = 3/(b+c-2a)
b1 = 1/(d - b) = 1/((a+b+c)/3 - b) = 3/(a+c-2b)
c1 = 1/(d - c) = 1/((a+b+c)/3 - c) = 3/(a+b-2c)
Тогда a1 > 0; b1 < 0; c1 > 0.
Поэтому b1 = |b1| = 3/(2b-a-c)
2 шаг:
a2 = 1/|a1| = |(b+c-2a)|/3 = (b+c-2a)/3
b2 = 1/|b1| = |(a+c-2b)|/3 = (2b-a-c)/3
c2 = 1/|c1| = |(a+b-2c)|/3 = (a+b-2c)/3
a2 + b2 + c2 = (b+c-2a)/3 + (2b-a-c)/3 + (a+b-2c)/3 =
= b/3 + c/3 - 2a/3 + 2b/3 - a/3 - c/3 + a/3 + b/3 - 2c/3 =
= (-2a/3 - a/3 + a/3) + (b/3 + 2b/3 + b/3) + (c/3 - c/3 - 2c/3) =
= -2a/3 + 4b/3 - 2c/3 = 2(2b - a - c)/3 = 2/b1 > 0
3 шаг.
Теперь эту сумму a2 + b2 + c2 надо умножить на a1, b1 и c1.
(a2 + b2 + c2)*a1 = (2/b1) * a1 = 2a1/b1
(a2 + b2 + c2)*b1 = (2/b1) * b1 = 2 - вот она появилась!
(a2 + b2 + c2)*c1 = (2/b1) * c1 = 2c1/b1

Answer 2

The Pattern and Role of Arithmetic Mean in the Given Process

The pattern you described involves performing a series of calculations with three given numbers: 1.43, 4.50, and 7.40, and observing that the result always includes the number 2.00. You also mentioned the role of the arithmetic mean in this process. Let's break down the steps and explain the pattern and the role of the arithmetic mean.

1. Step 1: Calculating the Arithmetic Mean - The arithmetic mean, also known as the average, is calculated by summing up all the numbers and dividing the sum by the total count of numbers. - In this case, the arithmetic mean of the three numbers (1.43, 4.50, and 7.40) is (1.43 + 4.50 + 7.40) / 3 = 4.4433333333333333.

2. Step 2: Calculating the Inverse of the Arithmetic Mean - The next step involves calculating the inverse of the arithmetic mean obtained in Step 1. - The inverse of a number x is obtained by dividing 1 by x. - In this case, the inverse of the arithmetic mean (4.4433333333333333) is 1 / 4.4433333333333333 = 0.331858...

3. Step 3: Applying the Inverse to the Given Numbers - The inverse obtained in Step 2 is then applied to each of the three given numbers: 1.43, 4.50, and 7.40. - For example, applying the inverse (0.331858...) to 1.43 gives 1 / 1.43 = 0.331858... - Similarly, applying the inverse to 4.50 gives 1 / 4.50 = 0.222222... and applying it to 7.40 gives 1 / 7.40 = 0.135135...

4. Step 4: Multiplying the Results with the Inverse - The next step involves multiplying the results obtained in Step 3 with the inverse (0.331858...) obtained in Step 2. - For example, multiplying 0.331858... with 0.331858... gives 0.331858... * 0.331858... = 0.109999... (approximately 0.11). - Similarly, multiplying 0.222222... with 0.331858... gives 0.073333... (approximately 0.07), and multiplying 0.135135... with 0.331858... gives 0.044999... (approximately 0.04).

5. Step 5: Rounding the Results - Finally, the results obtained in Step 4 are rounded to two decimal places. - For example, rounding 0.11 gives 0.11, rounding 0.07 gives 0.07, and rounding 0.04 gives 0.04.

6. The Role of Arithmetic Mean - The arithmetic mean plays a role in this process by providing a reference value for the calculations. - The inverse of the arithmetic mean is used to transform the given numbers, and multiplying the transformed numbers with the inverse brings back the number 2.00. - The presence of the number 2.00 in the final results is a consequence of the calculations involving the arithmetic mean and its inverse.

In summary, the pattern you observed in the given process is a result of applying the inverse of the arithmetic mean to the given numbers and then multiplying the transformed numbers with the inverse. The arithmetic mean serves as a reference value in this process, and the number 2.00 appears consistently in the final results.