For what values of the parameter k are the three values of x such that 5^1+x+5^1-x, (k)/(2), 25^

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For what values of the parameter k are the three values of x...

For what values of the parameter $\mathrm{k}$ are the three values of $\mathrm{x}$ such that $5^{1+\mathrm{x}}+5^{1-\mathrm{x}}, \frac{\mathrm{k}}{2}, 25^{\mathrm{x}}+25^{-\mathrm{x}}$ are three successive terms of an AP

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Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence where the difference between any two consecutive terms is constant. A key property is that the middle term is the average of the first and third terms, which is central to setting up equations when given terms expressed in different forms.

Exponential Functions and Properties of Exponents

Exponential functions are expressions where the variable appears in the exponent. The manipulation of these functions often relies on the properties of exponents, such as rewriting 25^x as (5^2)^x = 5^(2x) and using the rule that a^(m+n) = a^m * a^n. This allows for rewriting terms with a common base, facilitating simplification and comparison.

Algebraic Manipulation and Transformation

Algebraic manipulation involves rearranging, factoring, and simplifying expressions to derive relationships or solve equations. When dealing with exponential terms and their sums, techniques such as substitution and applying identities (for example, expressing symmetric sums in terms of squares) are vital to establish the necessary equations.

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