For A.M.’s between $a$ and $b$: $A_1=\dfrac{2a+b}{3},\quad A_2=\dfrac{a+2b}{3}$ So, $A_1+A_2=\dfrac{2a+b+a+2b}{3}=a+b$ For G.M.’s between $a$ and $b$: $G_1=\sqrt[3]{a^2b},\quad G_2=\sqrt[3]{ab^2}$ So, $G_1G_2=\sqrt[3]{a^3b^3}=ab$ Hence, $\dfrac{A_1+A_2}{G_1G_2}=\dfrac{a+b}{ab}$