In an infinite geometric progression with positive terms and with a common ratio |r|<1, the sum of the first three terms is (26/3) and the sum of the entire progression is 9. Determine the progression.

Find the first term and common ratio

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To find the first term and the common ratio of an infinite geometric progression, we can use the formulas:

Sum of first n terms: S_n = a(1 - r^n) / (1 - r),
Sum of infinite terms: S = a / (1 - r),

where S_n is the sum of the first n terms, a is the first term, r is the common ratio.

Given that the sum of the first three terms is (26/3) and the sum of the entire progression is 9, we can set up the following equations:

(26/3) = a(1 - r^3) / (1 - r), (equation 1)
9 = a / (1 - r). (equation 2)

Now, let's solve these equations to find the values of a and r:

First, we can rearrange equation 2 to express a in terms of r:

a = 9(1 - r). (equation 3)

Next, substitute equation 3 into equation 1:

(26/3) = (9(1 - r))(1 - r^3) / (1 - r).

Multiply both sides of the equation by (1 - r) to eliminate the denominator:

(26/3)(1 - r) = 9(1 - r)(1 - r^3).

Expand and simplify the equation:

26 - (26/3)r = 9 - 9r - 9r + 9r^4.

Combine like terms:

26 - (26/3)r = 9(1 + r + r^2 + r^3).

Multiply both sides of the equation by 3 to eliminate the denominator:

78 - 26r = 27(1 + r + r^2 + r^3).

Rearrange the equation:

27 + 27r + 27r^2 + 27r^3 - 26r = 78.

Combine like terms:

27r^3 + 27r^2 + r - 51 = 0.

Now, we need to find the roots of this cubic equation. Unfortunately, there is no general formula to find the exact roots of a third-degree polynomial. However, we can use numerical methods such as factoring or a numerical approximation method like Newton's method to find an approximate solution.

Once we find the value(s) of r, we can substitute them back into equation 3 to find the corresponding values of a.

Note: The process to solve a cubic equation numerically is beyond the scope of a simple explanation. It involves iterative calculations and can be done using mathematical software or calculators.