Find the Domain (5^(n+2)-35*5^(n-1))/(18(5^(n+1)))

Problem

(5(n+2)−35⋅5(n−1))/(18⋅5(n+1))

Solution

1. Identify the type of expression. This is a rational expression where the variable n appears in the exponents of the terms.

2. Determine the condition for the domain. For a rational expression to be defined, the denominator must not be equal to zero.

3. Set up the inequality for the denominator. We must solve for n such that 18⋅5(n+1)≠0

4. Analyze the exponential term. The base 5 is a positive constant, and an exponential function of the form ax (where a>0 is always strictly positive for all real numbers.

5. Conclude that since 5(n+1)>0 for all n the product 18⋅5(n+1) is never zero.

6. State the domain. Since there are no values of n that make the denominator zero and the exponents are defined for all real numbers, the domain is all real numbers.

Final Answer

Domain: *(−∞,∞)

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