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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.


Solution

 

As they are in AP, we have

a = 51+x + 51-x + 25x + 25-x.

So say 5x = t, then, t2 = 25x

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.

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