If the continued product of three numbers in G.P. is 216 and the sum of their products in pair is 156. Find the numbers.
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To solve this problem, let's first understand what a geometric progression (G.P.) is. In a G.P., each term is obtained by multiplying the preceding term by a constant factor called the common ratio (denoted by 'r').
Let's assume the three numbers in the geometric progression are a/r, a, and ar, where 'a' is the first term and 'r' is the common ratio.
Given that the continued product of the three numbers is 216, we can write the equation as follows: (a/r) * a * (ar) = 216.
Simplifying the equation, we get: a^3 * r^3 = 216.
Next, we are given that the sum of the products of the numbers taken in pairs is 156. The pairs of numbers are (a/r, a), (a/r, ar), and (a, ar).
Let's express these pair-wise products mathematically:
(a/r * a) + (a/r * ar) + (a * ar) = 156.
Simplifying further, we get: a^2/r + a^2 + a^2 * r = 156.
Now, we have a system of two equations:
1) a^3 * r^3 = 216
2) a^2/r + a^2 + a^2 * r = 156.
To solve this system of equations, we can use the method of substitution or elimination.
Let's solve it using the substitution method.
From the first equation, we can rewrite it as: a^3 = 216/r^3 and solve for a^3.
Substituting this value of a^3 in the second equation, we get:
(216/r^3) + (216/r) + (216 * r) = 156.
We can simplify this equation further, getting rid of the denominators by multiplying through by r^3:
216 + 216r^2 + 216r^4 = 156r^3.
Rearranging this equation, we have a polynomial equation:
216r^4 - 156r^3 + 216r^2 - 216 = 0.
Now, we can solve this equation to find the values of 'r'. Once we have 'r', we can substitute it back into either of the original equations to solve for 'a'.
Note: The equation is a quartic equation, and solving it might involve factorization, use of the Rational Root Theorem, or numerical methods (such as using a graphing calculator or software).