Equation in Progressions

The given numbers are $a,b$ . For two numbers $a,b : \space A = \dfrac{a + b}{2}, \space G^2 = ab, \space H = \dfrac{2ab}{a + b}$

Harmonic Mean $HM = \dfrac{2ab}{a + b} = 4 \implies 2ab = 4(a + b) \implies ab = 2(a + b)$

$2A + G^2 = 27 \implies 2 \times \dfrac{a + b}{2} + ab = 27$

$\implies (a + b) + 2(a + b) = 27$ (since $ab = 2(a + b)$

$\implies (a + b) = 9$

Now, from algebra we know that $(a + b)^2 - (a - b)^2 = 4ab$

$\implies (a - b)^2 = (9)^2 - 2 \times 4(a + b) = 81 - 72$ (since $2ab = 4(a + b)$ )

$\implies (a - b) = \pm 3$

If $a - b = +3$ by solving $a + b = 9 \space and \space a - b = +3$ we get :

$\implies a = 6, b = 3$

If $a - b = -3$ by solving $a + b = 9 \space and \space a - b = -3$ we get :

$\implies a = 3, b = 6$

$\color{#20A900}\text{Therefore, the required numbers are 3 and 6}$

Given that harmonic mean:

$\begin{aligned} H & = 4 \\ \frac 2{\frac 1a + \frac 1b} & = 4 \\ \frac {\color{#3D99F6}2\color{#D61F06}ab}{\color{#3D99F6}a+b} & = 4 & \small {\color{#3D99F6} A = \frac {a+b}2} \text{ and }\color{#D61F06} G = \sqrt{ab} \\ \frac {\color{#D61F06}G^2}{\color{#3D99F6}A} & = 4 \\ \implies G^2 & = 4A \end{aligned}$

Therefore,

$\begin{aligned} 2A + \color{#D61F06}G^2 & = 27 \\ 2A + \color{#D61F06}4A & = 27 \\ 6A & = 27 \\ \implies A & = \frac 92 & \small \color{#3D99F6} \implies a+b = 9 \implies G^2 =ab = 4A = 18 \end{aligned}$

Then we have:

$\begin{aligned} a+b & = 9 & \small \color{#3D99F6} \text{Multiply both sides by }a \\ a^2 + \color{#3D99F6} ab & = 9a & \small \color{#3D99F6} \text{Note that }ab = 18 \\ a^2 - 9a + \color{#3D99F6} 18 & = 0 \\ (a-3)(a-6) & = 0 \end{aligned}$

$\implies \begin{cases} a = 3 & \implies b = 6 \\ a = 6 & \implies b = 3 \end{cases}$

Therefore, $a$ and $b$ are $\boxed{\text{3 and 6}}$ .

Harmonic mean is n/(1/a + 1/b) where n = number of numbers, in this case 2.

So 2/(1/a + 1/b) = 4.

Therefore, 2/4 = 1/a + 1/b = 1/2.

1/4 + 1/8 = 3/8

1/9 + 1/12 = 7/36

1/6 + 1/9 = 5/18

1/3 + 1/6 = 3/6 = 1/2 ✔️

As check Arithmetic mean A = (3+6)/2=4.5 and 2*A = 9.

Geometric mean G = sqrt(3*6) = sqrt(18) so G^2 = 18.

Sum = 27 ✔️