If A.M. between two numbers exceeds their G.M. by 2 and the G.M. exceeds their H.M. by $\frac{8}{5},$ then the two numbers are

A = G + 2 
$\begin{array}{l}\mathrm{G}=\mathrm{H}+\frac{8}{5};{\mathrm{G}}^2=\mathrm{AH}=(\mathrm{G}+2)\left(\mathrm{G}-\frac{8}{5}\right)\\ {\mathrm{G}}^2={\mathrm{G}}^2+\frac{2}{5}\mathrm{G}-\frac{16}{5}\end{array}$
G = 8; G² = 64 = ab
A = 10; a + b = 20; a = 4, b = 1
a = 4, b = 16 or a = 16, b = 4