The product of three numbers in geometric progression is 1, their sum is -2/3. Find the numbers.
To find the three numbers in geometric progression, we will assume the first number is "a", the common ratio is "r", and the third number is "ar^2".
Given that the product of the three numbers is 1, we can write the equation:
a * ar^2 = 1 ----- (equation 1)
Also, the sum of the three numbers is -2/3, which gives us another equation:
a + ar + ar^2 = -2/3 ----- (equation 2)
Now, let's solve these two equations to find the values of "a" and "r" which in turn will help us find the three numbers.
From equation 1, we have:
a * ar^2 = 1
Expanding this equation, we get:
a^3 * r^2 = 1
Taking the cube root of both sides:
a * r = 1^(1/3)
a * r = 1
Now, let's substitute this value of "a * r" in equation 2:
a + ar + ar^2 = -2/3
Substituting "a * r" with 1, we have:
1 + 1 + ar^2 = -2/3
Combining like terms, we get:
2 + ar^2 = -2/3
Subtracting 2 from both sides:
ar^2 = -8/3
Now, substitute the value of "ar" from the earlier derived equation into this equation:
(1)(r^2) = -8/3
Simplifying this equation:
r^2 = -8/3
Taking the square root of both sides (considering both positive and negative roots):
r = ±√(-8/3)
(r = ±(2√2)i/√3)
Since "r" represents the common ratio, it cannot have complex or imaginary values in a geometric progression. Hence, the only valid solution is when "r" is equal to zero.
Therefore, r = 0
Substituting this value of "r" in equation 1, we get:
a * 0 = 1
0 = 1
However, this is not possible, as it violates the rule of finding the three numbers in geometric progression.
Hence, there are no valid solutions for this problem.