Question

Prove that the numbers $\sqrt{2},\sqrt{3},\sqrt{5}$ cannot be any three term (not necessarily consecutive) of an A.P.

Answer 1

$\sqrt{2},\sqrt{3,}\sqrt{5}d=\sqrt{3}-\sqrt{2}\ne \sqrt{5}-\sqrt{3}r=\frac{\sqrt{3}}{2}\ne \frac{\sqrt{5}}{3}$

Since $d$ and $r$ are not same then the numbers $\sqrt{2},\sqrt{3,}\sqrt{5}$ can not satisfy the condition to form of an $AP$ or $GP$.