Question

At what values of parameter ‘a’ are there values of n such that the numbers :

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

• A. a ≤ 8
• B. a ≥ 8
• C. a ≥ 12
• D. a ≤ 12

Answer 1

Answer: (C)

a ≥ 12

Answer 2

For the given numbers to be in A.P.,

$2\left( \frac{a}{2} \right) = \left( 5^{1 + x} + 5^{1 - x} \right) + \left( 25^{x} + 25^{- x} \right)$

Let $5^{x} = k$

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

⇒ $a = 5\left( k + \frac{1}{k} \right) + \left( k^{2} + \frac{1}{k^{2}} \right)$

As the sum of a positive number and its reciprocal is always greater than or equal to 2,

$k + \frac{1}{k} \geq 2$ and $k^{2} + \frac{1}{k^{2}} \geq 2$;

Hence $a \geq 5(2) + 2$⇒ $a \geq$12