First slide

Arithmetic progression

Question

$If x \in R, the numbers (5^{1 + x} + 5^{1 - x}), \frac{a}{2}, (25^{x} + 25^{- x}) from an A.P then 'a' must lie in the imterval :$

Moderate

A

$[1, 5]$
B

$[2, 5]$
C

$[5, 12]$
D

$[12, \infty)$

Solution

$\begin{array}{l} a = 5^{1 + x} + 5^{1 - x} + 25^{x} + 25^{- x} \\ a = 5 (5^{x} + \frac{1}{5^{x}}) + {(5^{x} + \frac{1}{5^{x}})}^{2} - 2 \\ Let t = 5^{x} + \frac{1}{5^{x}} \\ a = 5 t + t^{2} - 2 \\ a = t^{2} + 5 t - 2; t \geq 2 \\ a \geq 4 + 10 - 2 \\ a \geq 12 \end{array}$

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