If the sum of three number in GP is 65 and their product is 3375, then numbers are
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If the sum of three number in GP is 65 and their product is 3375, then numbers are
Again and again I am telling I am daring on u for 50 points
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Answer:
a = 65, ar = 65 * ∛(3/13), and ar^2 = 65 * (∛(3/13))^2.
Step-by-step explanation:
Let's denote the three numbers in the geometric progression as a, ar, and ar^2, where "a" is the first term, "r" is the common ratio, and "ar^2" is the third term.
We are given two pieces of information:
The sum of the three numbers is 65: a + ar + ar^2 = 65.
The product of the three numbers is 3375: a * ar * ar^2 = 3375.
Now, we can solve for these values. First, we'll simplify the product equation:
a * ar * ar^2 = 3375
a^3 * r^3 = 3375
Now, let's find the prime factorization of 3375:
3375 = 3^3 * 5^3
So, a^3 * r^3 = 3^3 * 5^3
a^3 * r^3 = (3 * 5)^3
Now, we can see that a^3 * r^3 is equal to (3 * 5)^3, which means a^3 * r^3 = 15^3.
So, a^3 * r^3 = 15^3 = 3375
Now, we know that a^3 * r^3 = 3375. Since the product of the three numbers is 3375, we can see that a^3 * r^3 = 3375.
Now, let's consider the sum equation:
a + ar + ar^2 = 65
We can factor out an "a" from the left side:
a(1 + r + r^2) = 65
Now, we need to find the value of (1 + r + r^2). Let's call this value "S":
S = 1 + r + r^2
Now, we have two equations:
a^3 * r^3 = 3375
a * S = 65
First, let's find the values of a and r from the second equation:
a = 65 / S
Now, substitute this expression for "a" into the first equation:
(65 / S)^3 * r^3 = 3375
Now, we need to solve for "r." To do this, let's simplify the equation:
r^3 = (3375 * S^3) / 65^3
r^3 = (15^3 * S^3) / 5^3 * 13^3
r^3 = (15 * S / 5 * 13)^3
r^3 = (3 * S / 13)^3
r = ∛(3 * S / 13)
Now, we can find "S" by substituting this expression for "r" back into the equation for "a":
a = 65 / S
Now, we have expressions for "a" and "r" in terms of "S." We can find "S" by solving this system of equations. Once we have "S," we can find "a" and "r."
Let's now find "S" by substituting these expressions for "a" and "r" into the equation:
(65 / S) * S = 65
Now, solve for "S":
S = 1
Now that we have found "S" (S = 1), we can find "a" and "r":
a = 65 / S = 65 / 1 = 65
r = ∛(3 * S / 13) = ∛(3 * 1 / 13) = ∛(3/13)
So, the three numbers in the geometric progression are a = 65, ar = 65 * ∛(3/13), and ar^2 = 65 * (∛(3/13))^2.