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 phase "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.
add the even numbers between 35 and 39
To solve this problem, we can start by setting up some equations.
Let's assume that the common difference of the arithmetic progression is "d", and the first term of the arithmetic progression is "a". The five terms would then be:
a, a + d, a + 2d, a + 3d, a + 4d
We are given that the 1st, 2nd, and 5th terms form a geometric progression. Let's assume that the common ratio of the geometric progression is "r".
Then, we have the following equations:
(a + d) / a = (a + 4d) / (a + 3d) = r
We are also given that the product of these five numbers is 124 4/9. So we can set up another equation:
a * (a + d) * (a + 2d) * (a + 3d) * (a + 4d) = 124 4/9
To find the product of the three terms of the geometric progression, we need to solve for "r".
We can start by solving the equations to find the values of "a" and "d".
Using the first equation, we can solve for "d":
(a + d) / a = (a + 4d) / (a + 3d)
Cross-multiplying the fractions, we get:
(a + d)*(a + 3d) = (a + 4d)*a
a^2 + 4ad + 3d^2 = a^2 + 4ad
3d^2 = 0
d = 0
This means that "d" is zero, but since we are looking for distinct positive reals, this is not possible. So, we made a mistake somewhere.
Let's try another approach:
Since the first, second, and fifth terms form a geometric progression, we have the following equation:
(a + d) / a = (a + 4d) / (a + 3d) = r
We can rewrite this equation as:
(a + d) / a = (a + 3d + d) / (a + 3d) = r
Cross-multiplying the fractions, we get:
(a + d)*(a + 3d) = (a + 3d + d)*a
a^2 + 3ad + ad + 3d^2 = a^2 + 4ad
4ad - 4d^2 = ad
3ad = 4d^2
Simplifying the equation, we get:
3a = 4d
Now, let's substitute this value of "d" back into the equation for the product of the five numbers:
a * (a + d) * (a + 2d) * (a + 3d) * (a + 4d) = 124 4/9
Substituting 4d with 3a, we get:
a * (a + 3a) * (a + 6a) * (a + 9a) * (a + 12a) = 124 4/9
a * 4a * 7a * 10a * 13a = 124 4/9
3640a^5 = 124 4/9
Dividing both sides by 3640, we get:
a^5 = 124 4/9 / 3640
a^5 = 124/3640 + 4/(9*3640)
a^5 = 31/910 + 1/8190
a^5 = (31*10 + 1)/9100
a^5 = 311/9100
To find the value of "a", we can take the fifth root of both sides:
a = (311/9100)^(1/5) ≈ 0.374
Now, we have the value of "a". We can substitute this into the equations we derived earlier to find the value of "d" and "r".
Using d = (3a) / 4:
d = (3*0.374) / 4 ≈ 0.28
Using r = (a + d) / a:
r = (0.374 + 0.28) / 0.374 ≈ 1.924
Now that we have the values of "a", "d", and "r", we can find the product of the three terms of the geometric progression.
The three terms would be:
a, a + d, a + 4d
The product of these three terms is:
(a) * (a + d) * (a + 4d)
Substituting the values we found:
(0.374) * (0.374 + 0.28) * (0.374 + 4*0.28) ≈ 0.28
Therefore, the product of the three terms of the geometric progression is approximately 0.28.