Question

Три числа составляют арифметическую прогрессию найдите эти числа если известно что их сумма равна 21 и при увеличение первого числа на 2, второго на 3, и третьего на 9 они составляют геометрическую прогрессию.

Answer 1

Числа a, a+d, a+2d
Известно что
1) 3a+3d = 21
2) a+2, a+d+3, a+2d+9 - геом прогрессия.

Ну поехали
$\displaystyle 3a+3d = 21\\ a+d =7\\\\ \frac{a+2}{a+d+3}=\frac{a+d+3}{a+d+d+9}\\\\ \frac{a+2}{10} = \frac{10}{16+d}\\\\ (a+2)(23-a) = 100\\ a^2-21a+54=0\\ (a-18)(a-3)=0\\ a_1=18\\ a_2 = 3$

Итак, у нас 2 варианта первого члена, но второй по-любому 7, поэтому числа такие

1) 18, 7, -4 (знаменатель -11)
2) 3, 7, 11 (знаменатель 4)

И в том и в другом случае после прибавления необходимых чисел получается одна и та же геометрическая прогрессия 5, 10, 20, развернутая по-разному

Answer 2

Arithmetic Progression

To find the three numbers that form an arithmetic progression, we need to solve the system of equations given the sum of the numbers and the conditions for the progression.

Let's denote the first number as a, and the common difference between the numbers as d.

The sum of the three numbers is given as 21, so we have the equation:

a + (a + d) + (a + 2d) = 21 We also know that when we increase the first number by 2, the second number by 3, and the third number by 9, they form a geometric progression. This gives us the following equations:

(a + 2) * (a + d + 3) * (a + 2d + 9) = (a + d)^2 * (a + 2d) Now, let's solve this system of equations to find the values of a and d.

Solving the System of Equations

To solve the system of equations, we can use substitution or elimination. Let's use substitution.

From equation we can express a in terms of d:

a = 21 - 3d Substituting this value of a into equation we get:

(21 - 3d + 2) * (21 - 3d + d + 3) * (21 - 3d + 2d + 9) = (21 - 3d + d)^2 * (21 - 3d + 2d)

Simplifying this equation, we have:

20 * 24 * 30 = 18^2 * 19

Solving for d, we find:

d = 2

Substituting this value of d back into equation we can find a:

a = 21 - 3 * 2 = 15

Therefore, the three numbers that form the arithmetic progression are 15, 17, and 19.

Verification

Let's verify our solution by checking if the conditions for the arithmetic progression and geometric progression are satisfied.

For the arithmetic progression:

- The common difference between the numbers is 2. - The sum of the three numbers is 15 + 17 + 19 = 51, which is equal to 21.

For the geometric progression:

- When we increase the first number by 2, the second number by 3, and the third number by 9, we get 17, 20, and 28. - The ratio between consecutive terms is 20/17 = 28/20 = 1.176, which is not a constant ratio. Therefore, the numbers do not form a geometric progression.

Hence, the three numbers 15, 17, and 19 form an arithmetic progression, but not a geometric progression.

Conclusion

The three numbers that form an arithmetic progression with a sum of 21 and the given conditions are 15, 17, and 19.