First slide

Arithmetic progression

Question

If $5^{1 + x} + 5^{1 - x}, \frac{a}{2} and 25^{x} + 25^{- x}$ are three consecutive terms of an A.P., then the values of a are given by

Moderate

A

$a \geq 12$
B

$a > 12$
C

$a < 12$
D

$a \leq 12$

Solution

Since, $5^{1 + x} + 5^{1 - x}, \frac{a}{2}, 25^{x} + 25^{- x}$ are in A.P., we have
$2 \frac{a}{2} = 5^{1 + x} + 5^{1 - x} + 25^{x} + 25^{- x}$
Now put 5^x = t so that t > 0, we then have
$\begin{aligned} a = 5 t + \frac{5}{t} + t^{2} + \frac{1}{t^{2}} = (t^{2} + \frac{1}{t^{2}}) + 5 (t + \frac{1}{t}) \\ or a = {(t - \frac{1}{t})}^{2} + 2 + 5 [{(\sqrt{t} - \frac{1}{\sqrt{t}})}^{2} + 2] \\ = {(t - \frac{1}{t})}^{2} + 5 {(\sqrt{t} - \frac{1}{\sqrt{t}})}^{2} + 12 \geq 12 \end{aligned}$
Thus, values of a are given by the inequality a ≥ 12.

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