Explanation
Concept
Let x and y be the two numbers. The arithmetic mean A, geometric mean G and the harmonic mean H of x and y is given by,
A=2x+y
</span><br><spanstyle="font−size:14pt;">G2=xy
H=x+y2xy
Calculations:
Consider, the two numbers are x and y.
Given, the arithmetic mean and geometric mean of the x and y is A and G.
</span><br><spanstyle="font−size:14pt;">A=2x+y…(1)
G2=xy…(2)
The harmonic mean of two number x and y is 4.
</span><br><spanstyle="font−size:14pt;">x+y2xy=4
2xy=4(x+y)
</span><br><spanstyle="font−size:14pt;">xy=2(x+y)
⇒G2=4A(∵x+y=2A)
⇒G2=4A....(3)
Given, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G^2 = 27.
<br>⇒2A+G2=27
⇒6A=27
<br>⇒A=29
From equation (1), (2) and (3), we have
x + y = 9 and xy = 18
⇒x=6 and y=3
Hence, the harmonic mean of two number is 4, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G^2 = 27, then the two numbers are 6 and 3.
