Answer :

Final answer:

To find the long run value of the shifted geometric sequence, find the common ratio using the n-th term and its previous term. Calculate the first term by solving the given equation. Use the formula for the sum of an infinite geometric series, substituting the found values, to find the long run value.

Explanation:

To find the long run value of a shifted geometric sequence, we can first determine the common ratio of the sequence. In this case, the common ratio can be found by dividing the n-th term by its previous term. So, 0.75 is the common ratio in this sequence. Next, we can find the first term (u(0)) by plugging in the equation u(n) = 0.75u(n-1) + 25 with n=1. Therefore, u(0) = 0.75u(0) + 25, and solving this equation gives us u(0) = 100. Now that we have the common ratio and the first term, we can use the formula for the sum of an infinite geometric series to find the long run value. The formula is S = a / (1 - r), where S represents the sum, a is the first term, and r is the common ratio. Using the given values, we have S = 100 / (1 - 0.75), which simplifies to S = 100 / 0.25, and finally S = 400.

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