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Question

At what values of the parameter $$a$$ are there values of $$x$$ such that the numbers $$\displaystyle\, 5^{1 + x}\, +\, 5^{1 - x},  \frac{a}{2},   25^x\, +\, 25^{-x}$$ form an arithmetic progression?


Solution

$$5^{1+x}+5^{1-x},\dfrac{a}{2},25^{x}+25^{-x}$$ forms an $$A.P$$
$$\implies a=5^{1+x}+5^{1-x}+5^{2{x}}+5^{-2{x}}$$
$$AM\ge GM$$
$$\dfrac{5^{1+x}+5^{1-x}+5^{2 x}+5^{-2 x}}{4}\ge \sqrt[4]{5^{1+x+1-x+2{x}-2{x}}}$$
$$\implies a\ge 4\sqrt{5}$$

Mathematics

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