Question

At what values of parameter 'a' are there values of n such that the numbers 5^(1+x)+5^(1-x),a/2, 25^(x)+25^(-x) form an AP.

Answer 1

As they are in AP, we have

a = 5^1+x + 5^1-x + 25^x + 25^-x.

So say 5^x = t, then, t² = 25^x

a = 5t + 5/t + t² + 1/t².

Now 5t+5/t ≥ 10 --------- (AM-GM Inequality)

and t²+1/t² ≥ 2 --------- (AM-GM Inequality)

So a ≥ 12, are the values that "a" can take.