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Saturday, November 1, 2008

Q) If the arithmetic mean of two numbers 'a' and 'b' is 'n' times their harmonic mean, find a/b in terms of 'n'.

A) AM = n. HM
(a+b) / 2 = n. [2ab/(a+b)]
a^2 + b^2 + 2ab = 4*n*ab
(a+b)^2 = 4*n*ab
(a/b + 1)^2 = 4*n*a/b
Let a/b = x
(x + 1)^2 = 4nx
x^2 + 1 + 2x - 4nx = 0
x^2 + x(2-4n) + 1 = 0
x = 4n-2 +- sqrt(16n^2 + 4 - 16n -4) / 2
x = 4n-2 +- 4.sqrt(n^2 - n) / 2
x = 2n - 1 +- 2.sqrt (n^2 - n)
OR
a/b = 2n - 1 +- 2.sqrt (n^2 - n)


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1 comment:

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