Question

At what values of the parameter $a$ are there values of $x$ such that the numbers $5^{1+x}+5^{1-x},\frac{a}{2},{25}^x+{25}^{-x}$ form an arithmetic progression?

Answer 1

$5^{1+x}+5^{1-x},\frac{a}{2},{25}^x+{25}^{-x}$ forms an $A.P$

$⟹a=5^{1+x}+5^{1-x}+5^{2x}+5^{-2x}$

$AM\ge GM$

$\frac{5^{1+x}+5^{1-x}+5^{2x}+5^{-2x}}{4}\ge \sqrt[4]{45^{1+x+1-x+2x-2x}}$

$⟹a\ge 4\sqrt{5}$