The Harmonic mean of two numbers is 4, their Arithmetic mean $A$ and Geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the sum of those numbers.

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Solution

If the two numbers be $x$ and $y, A = \frac{x + y}{2}, G = \sqrt{x y}, H = \frac{2 x y}{x + y}$
Thus, $x y = 2 x + 2 y$ since $H = 4$
Now, $2 A + G^{2} = 27$
$\Rightarrow x + y + x y = 27$
$\Rightarrow x + y = 9$ and $x y = 18$
Thus, the numbers are $6, 3$

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The Harmonic mean of two numbers is 4, their Arithmetic mean A and Geometric mean G satisfy the relation 2A+G2=27. Find the sum of those numbers.

If the two numbers be x and y,A=x+y2,G=√xy,H=2xyx+yThus, xy=2x+2y since H=4Now,2A+G2=27⇒x+y+xy=27⇒x+y=9 and xy=18Thus, the numbers are 6,3

The Harmonic mean of two numbers is 4, their Arithmetic mean $A$ and Geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the sum of those numbers.

If the two numbers be $x$ and $y, A = \frac{x + y}{2}, G = \sqrt{x y}, H = \frac{2 x y}{x + y}$
Thus, $x y = 2 x + 2 y$ since $H = 4$
Now, $2 A + G^{2} = 27$
$\Rightarrow x + y + x y = 27$
$\Rightarrow x + y = 9$ and $x y = 18$
Thus, the numbers are $6, 3$