Question

Даны четыре последовательных члена возрастающей геометрической прогрессии. Сумма двух крайних членов равна 14, а сумма двух средних равна 6. Определи четвертый член этой прогрессии.

Answer 1

Пусть b - первый член, q - знаменатель геометрической прогресии, q > 1
По условию
b + b*q^3 = 14
b*q + b * q^2 = 6

q - 1 + 1/q = 7/3
q - 10/3 + 1/q = 0
3q^2 - 10q + 3 = 0
q1 = 1/3, q2 = 3
По условию q > 1, т.е. q = 3, тогда b = 1/2, тогда четвёртый член равен 1/2 * 3^3 = 27/2

Answer 2

I will answer your question in detail. You are given four consecutive terms of an increasing geometric progression. The sum of the two extreme terms is equal to 14, and the sum of the two middle terms is equal to 6. You need to determine the fourth term of this progression.

To solve this problem, we can use the following facts and formulas about geometric progressions :

- A geometric progression is a sequence of numbers , , , (terms of the progression), in which the first term is not zero, and each subsequent term, starting from the second, is obtained by multiplying the previous term by a fixed non-zero number for this sequence (the common ratio of the progression). - Any term of a geometric progression can be calculated by the formula , where is the first term, is the common ratio, and is the term number. - The sum of the first terms of a geometric progression can be calculated by the formula , where is the first term, is the common ratio, and is the number of terms.

Let us denote the four consecutive terms of the given progression by , , , and . Then we have the following system of equations:

- , since the sum of the two extreme terms is equal to 14. - , since the sum of the two middle terms is equal to 6.

We can simplify the first equation by dividing both sides by , and get:

- .

We can also simplify the second equation by dividing both sides by , and get:

- .

Now we have a system of two equations with two unknowns, and . We can solve it by substitution or elimination method. For example, we can express from the first equation and substitute it into the second equation:

- . - .

We can simplify this equation by multiplying both sides by , and get:

- .

This is a quadratic equation, which can be solved by the formula:

- , where , , and .

In our case, , , and . Substituting these values into the formula, we get:

- .

We can discard the negative root, since the common ratio of an increasing geometric progression must be positive. Therefore, we have:

- .

Now we can find by substituting into the first equation:

- .

Finally, we can find the fourth term of the progression by using the formula :

- .

Therefore, the answer is: the fourth term of the given geometric progression is 9.

Answer 3

Problem Analysis

We are given four consecutive terms of an increasing geometric progression. The sum of the two outer terms is 14, and the sum of the two middle terms is 6. We need to determine the fourth term of this progression.

Solution

Let's assume the four consecutive terms of the geometric progression are a, ar, ar^2, and ar^3, where a is the first term and r is the common ratio.

We are given that the sum of the two outer terms is 14, so we have the equation: a + ar^3 = 14

We are also given that the sum of the two middle terms is 6, so we have the equation: ar + ar^2 = 6

To solve these equations, we can use substitution. We can solve the second equation for a and substitute it into the first equation: a = 6 / (r + r^2)

Substituting this value of a into the first equation, we get: (6 / (r + r^2)) + (6 / (r + r^2)) * r^3 = 14

Simplifying this equation, we get: 6 + 6r^3 = 14(r + r^2)

Expanding and rearranging the equation, we get: 14r^2 + 14r - 6r^3 - 6 = 0

Now, we can solve this equation to find the value of r. Once we have the value of r, we can substitute it back into the equation a = 6 / (r + r^2) to find the value of a. Finally, we can calculate the fourth term of the geometric progression by multiplying the third term (ar^2) by the common ratio (r).

Let's solve the equation to find the value of r.

Solution Steps

1. Start with the equation: 14r^2 + 14r - 6r^3 - 6 = 0. 2. Simplify the equation: -6r^3 + 14r^2 + 14r - 6 = 0. 3. Factor out -2: -2(r^3 - 7r^2 - 7r + 3) = 0. 4. Factor the cubic equation: (r - 1)(r + 1)(r - 3) = 0. 5. Solve for r: r = 1, r = -1, r = 3. 6. Substitute each value of r back into the equation a = 6 / (r + r^2) to find the corresponding values of a. 7. Calculate the fourth term of the geometric progression by multiplying the third term (ar^2) by the common ratio (r).

Let's calculate the values of r and a using the steps outlined above.

Calculation

1. For r = 1: - a = 6 / (1 + 1^2) = 6 / 2 = 3 - Fourth term = ar^3 = 3 * 1^3 = 3

2. For r = -1: - a = 6 / (-1 + (-1)^2) = 6 / 0 = undefined (division by zero) - Fourth term is not defined for this value of r.

3. For r = 3: - a = 6 / (3 + 3^2) = 6 / 12 = 0.5 - Fourth term = ar^3 = 0.5 * 3^3 = 0.5 * 27 = 13.5

Answer

The fourth term of the geometric progression is 3 when the common ratio is 1, and it is 13.5 when the common ratio is 3.

Note: The fourth term is not defined for a common ratio of -1.