Question

Найдите пятый член геометрической прогрессии, состоящей из восьми членов , если сумма её членов с чётными номерами равна 1360, а с нечётными 680. Помогите пожалуйста))

Answer 1

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Answer 2

Finding the Fifth Term of a Geometric Progression

To find the fifth term of a geometric progression, we need to use the given information about the sum of the terms with even and odd indices.

Let's denote the first term of the geometric progression as a and the common ratio as r. Since the geometric progression consists of eight terms, the eighth term can be expressed as:

a * r^7

We are given that the sum of the terms with even indices is 1360, which can be expressed as:

a + a * r^2 + a * r^4 + a * r^6 = 1360 Similarly, the sum of the terms with odd indices is 680, which can be expressed as:

a * r + a * r^3 + a * r^5 + a * r^7 = 680 To solve these two equations, we can use algebraic manipulation. Let's divide equation by equation:

(a * r + a * r^3 + a * r^5 + a * r^7) / (a + a * r^2 + a * r^4 + a * r^6) = 680 / 1360

Simplifying this equation gives:

r / (1 + r^2 + r^4 + r^6) = 1/2

Now, let's solve for r:

r = (1/2) * (1 + r^2 + r^4 + r^6)

This equation can be rearranged as:

2r = 1 + r^2 + r^4 + r^6

Now, let's substitute r with x to simplify the equation:

2x = 1 + x^2 + x^4 + x^6

This equation can be rewritten as a polynomial equation:

x^6 + x^4 + x^2 - 2x + 1 = 0

By solving this polynomial equation, we can find the value of x. However, finding the exact value of x may require complex calculations. Therefore, we can use numerical methods or approximation techniques to find an approximate value for x.

Once we have the value of x, we can substitute it back into equation or to find the value of a. Finally, we can calculate the fifth term of the geometric progression using the formula:

a * r^4

Please note that due to the complexity of the calculations involved, it may be more efficient to use numerical methods or approximation techniques to find the exact value of the fifth term.