Ta có $\mathrm{\frac 1{\sqrt{U_1}+\sqrt{U_2}}=\frac{\sqrt U_2-\sqrt{U_1}}{U_2-U_1}=\frac{\sqrt{U_2}-\sqrt{U_1}}{d}}$
$\mathrm{\frac{1}{\sqrt{U_2}+\sqrt{U_3}}=\frac{\sqrt{U_3}-\sqrt{U_2}}{d}}$
$....$
$\mathrm{\frac{1}{\sqrt{U_{n-1}}+\sqrt{U_n}}=\frac{\sqrt{U_n}-\sqrt{U_{n-1}}}{d}}$
Lấy tổng lại:
$\mathrm{\Rightarrow VT=\frac{\sqrt{U_n}-\sqrt{U_{1}}}{d}=\frac{(\sqrt{U_n}-\sqrt{U_{1}})(\sqrt{U_n}+\sqrt{U_{1}})}{d(\sqrt{U_n}+\sqrt{U_{1}})}=\frac{(n-1).d}{(\sqrt{U_n}+\sqrt{U_{1}}).d}=VP}$