The sum of three numbers in g.p. is 21 and the sum of their squares is 189. Find the numbers.
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To find the numbers, we need to solve the given problem step by step.
Let's assume the three numbers in geometric progression (g.p.) are a/r, a, and ar, where 'a' is the first term and 'r' is the common ratio.
According to the problem, the sum of the three numbers is 21, so we can write the equation as:
(a/r) + a + (ar) = 21 ----(1)
Similarly, the sum of their squares is 189, so the equation becomes:
(a/r)^2 + a^2 + (ar)^2 = 189 ----(2)
Let's simplify the equations further.
From equation (1), we can rewrite it as:
(a/r) + a + (ar) = 21
a(1/r + 1 + r) = 21 ----(3)
From equation (2), we can rewrite it as:
(a/r)^2 + a^2 + (ar)^2 = 189
a^2(1/r^2 + 1 + r^2) = 189 ----(4)
Now, let's solve equations (3) and (4) simultaneously.
From equation (3), we can express 'a' in terms of 'r':
a = 21/(1/r + 1 + r) ----(5)
Now substitute equation (5) into equation (4):
(21/(1/r + 1 + r))^2 (1/r^2 + 1 + r^2) = 189
Let's simplify this equation further.
Multiply both sides of the equation by (1/r + 1 + r)^2 to eliminate the fraction:
21^2 (1/r^2 + 1 + r^2) = 189 (1/r + 1 + r)^2
441 (1 + r^2 + r^4 + 2/r + 2/r^2 + 2r) = 189 (1 + 2/r + 2r + 4 + 2r^2 + 2/r)
Expanding both sides of the equation:
441 + 441r^2 + 441r^4 + 882/r + 882/r^2 + 882r = 189 + 378/r + 378r + 756 + 378r^2 + 378/r
Grouping like terms:
441r^4 + 441/r^2 + 882r^2 + 882/r + 441r^2 + 882r - 378r^2 - 378/r - 378 - 378r - 756 = 0
Simplifying further:
441r^4 + 63r^2 - 114r - 504 = 0
Now we have a quartic equation in terms of 'r'. We can solve this equation using numerical methods or factoring techniques.
Once we find the values of 'r', we can substitute them back into equation (5) to find the corresponding values of 'a'.