Final answer:
By simplifying the given algebraic equation, we find that k = -25. However, as all provided choices are positive, there seems to be a discrepancy, which suggests a possible typo or mistake in the question or its options.
Step-by-step explanation:
We are given the algebraic equation ( 5^n + 2 - 5^n - 25(5^n - 1) = k · (5^n) ) and we are asked to find the value of the constant (k). To solve this, we'll simplify the terms on the left side of the equation by canceling out like terms and factoring out 5^n where possible. Let's start by simplifying:
(5^n + 2 - 5^n - 25(5^n - 1))
= (5^n - 5^n) + 2 - 25 · 5^n + 25
= 0 + 2 - 25 · 5^n + 25
= 2 + 25 - 25 · 5^n
=27 - 25 · 5^n
Now, we set this equal to k · 5^n:
27 - 25 · 5^n = k · 5^n
Since the terms involving 5^n must cancel out for equality, k must be equal to -25:
k = -25
However, since the choices a) through d) are all positive and do not include -25, there might be a typo or mistake in the given answers or the equation. Assuming this is the case, our result does not match the provided choices, which means the question may need to be re-evaluated for possible errors.