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For real x, the three quantities 5^(1+x) + 5^(1-x), a/2, and 25^x + 25^(-x) form an arithmetic progression. Determine the range of values that 'a' must belong to.

1. [1, 5]
2. [2, 5]
3. [5, 12]
4. [12, infinity)

Correct answer: [12, infinity)

Solution

Setting a/2 as the middle term: 2*(a/2) = (5^(1+x)+5^(1-x)) + (25^x+25^(-x)). With t = 5^x+5^(-x) >= 2, we get a = 5t + t² - 2 = t² + 5t - 2. The minimum at t=2 gives a = 4 + 10 - 2 = 12, so a is in [12, infinity).

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