Question

The harmonic mean of 2 numbers is 4. Their arithmetic mean A and geometric mean G satisfy the relation 2A + $G^2$ = 27. The sum of the numbers is __

Answer 1

Let the numbers be a & b

H = 4

$G^2$ = AH = 4A -------------(1)

Given 2A + $G^2$ = 27 -----------(2)

From (1) & (2)

6A = 27 $\Rightarrow$ A = $\frac{9}{2}$

$G^2$ = 4 $\times$ $\frac{9}{2}$ = 18

If we know A and G, the 2 numbers are given by,

a, b = A $\pm$ $\sqrt{A^2-G^2}$

= $\frac{9}{2}$ $\pm$ $\sqrt{\frac{81}{4}-18}$

= $\frac{9}{2}$ $\pm$ $\frac{3}{2}$ = 6, 3 (or) 3, 6

Therefore, the sum of the numbers = 9