For two numbers the means satisfy $\text{GM}^2 = \text{AM} \times \text{HM}$. So $\text{HM} = \dfrac{\text{GM}^2}{\text{AM}} = \dfrac{4^2}{5} = \dfrac{16}{5} = 3.2$. (This matches finding the numbers 2 and 8 and computing their HM $= \frac{2\cdot 2\cdot …

Step 1 — Use the AM–GM–HM identity

For two positive numbers,

Step 2 — Solve for HM

Step 3 — Cross-check with the actual numbers

AM $=5$, GM $=4$ give the numbers 2 and 8, whose HM is

$\frac{2ab}{a+b} = \frac{2\times 2\times 8}{2+8} = \frac{32}{10} = 3.2\;(\checkmark)$ …