the sum of the three numbers in GP IS 35 and their product is 1000. find the numbers
a+ar+ar^2 = 35
a*ar*ar^2 = 1000
now just solve for a and r.
To find the numbers, we need to apply the concept of a geometric progression (GP) and use a system of equations.
Let's denote the three numbers in the GP as a, ar, and ar^2 (where a is the first term and r is the common ratio).
Given that the sum of the three numbers is 35, we can write the equation as:
a + ar + ar^2 = 35 ----(Equation 1)
We are also given that the product of the three numbers is 1000, so we can write the equation as:
a * ar * ar^2 = 1000
a^3 * r^3 = 1000
Now, we can rewrite Equation 1 using the value of a from Equation 2:
(a^3 * r^3) / a^2 = 1000
ar^3 = 1000
Substituting the value of ar^3 into Equation 1:
a + ar + (ar^3 / r^2) = 35
a + ar + ar = 35
2ar + a = 35
a(2r + 1) = 35
We can now find the value of a by analyzing the factors of 35:
For a to be an integer, the possible values for a are 1, 5, 7, and 35.
Now, substitute each value of a into the equation (2r + 1) = 35 / a to find the corresponding values of r:
1. For a = 1: (2r + 1) = 35 / 1 => 2r + 1 = 35 => 2r = 34 => r = 17
2. For a = 5: (2r + 1) = 35 / 5 => 2r + 1 = 7 => 2r = 6 => r = 3
3. For a = 7: (2r + 1) = 35 / 7 => 2r + 1 = 5 => 2r = 4 => r = 2
4. For a = 35: (2r + 1) = 35 / 35 => 2r + 1 = 1 => 2r = 0 => r = 0
From the above computations, we can see that r = 3 and a = 5 produce integer values for a and r. Thus, the three numbers in the geometric progression are:
a = 5
ar = 5 * 3 = 15
ar^2 = 5 * 3^2 = 45
So, the three numbers are 5, 15, and 45.