Consider, the two numbers are x and y.
Given, the arithmetic mean and geometric mean of the x and y is A and G.
⇒A= .......(1)
⇒G2=xy ......(2)
The harmonic mean of two number
x and
y is
4. ⇒=4 ⇒2xy=4(x+y) ⇒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+G2=27. ⇒2A+G2=27 ⇒6A=27 ⇒A= 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+G2=27 , then the two numbers are 6 and 3.