Final answer:
To find the explicit rule for the given sequence of values, we can use exponent properties and algebraic manipulation. By analyzing the given values and using the exponent properties, the explicit rule for the sequence is determined to be a = (85/25)^(1/6).
Step-by-step explanation:
To find the explicit rule for the sequence of values given, we need to determine the pattern or relationship between the terms. Let's analyze the given values:
a⁵ = 25 and a¹¹ = 85
To solve this, we can use the exponent properties. Since a⁵ = 25, we can rewrite it as (a^1) * (a^4) = 25. Similarly, we can rewrite a¹¹ = 85 as (a^1) * (a^10) = 85. From these equations, we can see that a^1 is common in both expressions. So, we can write a^1 * (a^4) = 25 and a^1 * (a^10) = 85.
To find the explicit rule, divide the second equation by the first equation: (a^1 * (a^10))/(a^1 * (a^4)) = 85/25. This simplifies to: a^6 = 85/25. Now, raise both sides to the power of 1/6 to solve for a: (a^6)^(1/6) = (85/25)^(1/6). This further simplifies to: a = (85/25)^(1/6). Therefore, the explicit rule for the sequence of values is a = (85/25)^(1/6).