5 distinct positive reals form an arithmetic progression. The 1st, 2nd and 5th term form a geometric progression. If the product of these 5 numbers is 124 4/9, what is the product of the 3 terms of the geometric progression?

Note: The phrase "form an arithmetic progression" means that the values are consecutive terms of an arithmetic progression. Similarly, "form a geometric progression" means that the values are consecutive terms of a geometric progression.

U5=a t(n-1)d

Let's denote the five terms of the arithmetic progression as a, a+d, a+2d, a+3d, and a+4d, where a is the first term and d is the common difference.

We are given that the first, second, and fifth terms form a geometric progression. This means that (a) * (a+d) * (a+4d) = 124 4/9.

Expanding this equation, we get:
a^3 + 5ad^2 + 4ad^2 + 4d^3 = 124 4/9.

Combining like terms, we have:
a^3 + 9ad^2 + 4d^3 = 124 4/9.

Now, let's consider the product of the three terms in the geometric progression: a * (a+d) * (a+4d).

Expanding this equation, we get:
a^3 + 5ad^2 + 4ad^2.

Comparing this equation with the previous one, we see that the product of the three terms in the geometric progression is equal to 124 4/9 - 4d^3.

To find the value of the product, we need to determine the value of d.

To solve this problem, let's break it down step by step.

Step 1: Let's assume that the common difference of the arithmetic progression is d, and the first term is a.

Step 2: Since the 1st, 2nd, and 5th terms form a geometric progression, let's denote the 1st term as a, the common ratio as r, and the 5th term as ar^3.

Step 3: Now, we can express the other terms in the arithmetic progression in terms of a and d. The 2nd term would be a + d, and the 5th term would be a + 4d.

Step 4: We can then set up equations using the given information. Since the product of the 5 numbers is 124 4/9, we can multiply all the terms together and set it equal to 124 4/9:
a * (a + d) * (a + 4d) * a * ar^3 = 124 4/9

Step 5: Simplifying the equation, we have:
a^2 * (a + d) * (a + 4d) * r^3 = 124 4/9

Step 6: Now, let's solve for the product of the terms in the geometric progression. We know that the product of the terms in a geometric progression is a * (a + d) * (a + 4d), since these are the terms involved in the geometric progression.

Step 7: To find the product, we need to solve for a and d. Unfortunately, the equation we have is not easily factorizable. However, we can use numeric methods or calculators to find the values of a, d, and r that satisfy the equation,

Step 8: Once we have the values of a, d, and r, we can substitute them into the expression for the product of the terms in the geometric progression, which is a * (a + d) * (a + 4d), to find the final answer.

Therefore, the product of the three terms in the geometric progression can be found by solving the equation and then calculating the expression a * (a + d) * (a + 4d) with the obtained values of a and d.