Question

For what values of the parameter a such that $5^{1+x}+5^{1-x},\frac{a}{2},{25}^x+{25}^{-x}$ are three consecutive terms of an A.P.?

Answer 1

As they are in A.P.

$a=5^{1+x}+5^{1-x}+{25}^x+{25}^{-x}$

So say $5^x=t$ then,

$a=5t+\frac{5}{t}+t^2+\frac{1}{t^2}$

Now, $5t+\frac{5}{t}\ge 10$ [A.M. - G.M. Inequality ]

and $t^2+\frac{1}{t^2}\ge 2$ [A.M. - G.M. Inequality ]

So, $a\ge 12$ are the values which $a$ can take.