Question

Rhonda started a business. Her business made $30,000 in profits the first year. Her annual profits have increased by an average of 5% each year since then.

(a) Write an iterative rule to model the sequence formed by the profits of Rhonda’s business each year.
(b) Use the rule to determine what the annual profits of Rhonda’s business can be predicted to be 15 years from the start of her business. Round your answer to the nearest dollar. Do not round until the end. Show your work.

Answer 1

(a) The iterative rule to model the sequence formed by the profits of Rhonda's business each year can be written as follows:

→ P(0) = $30,000 (initial profit)

→ P(n) = P(n-1) + 0.05 * P(n-1)

In this formula, P(0) represents the initial profit of $30,000, P(n) represents the profit in the nth year, and P(n-1) represents the profit in the previous year. The term 0.05 * P(n-1) represents the 5% increase in profits each year.

(b) To determine the annual profits of Rhonda's business 15 years from the start, we can use the iterative rule. Starting from the initial profit of $30,000, we can calculate the profit for each year using the formula mentioned in part (a).

Let's calculate the profits step-by-step:

→ Year 1: P(1) = P(0) + 0.05 * P(0) = $30,000 + 0.05 * $30,000 = $31,500

→ Year 2: P(2) = P(1) + 0.05 * P(1) = $31,500 + 0.05 * $31,500 = $33,075

→ Year 3: P(3) = P(2) + 0.05 * P(2) = $33,075 + 0.05 * $33,075 = $34,728.75

→ Year 15: P(15) = P(14) + 0.05 × P(14)

We can continue this process for 15 years, substituting the value of P(n-1) into the formula to find P(n). Finally, rounding the result to the nearest dollar will give us the predicted annual profit of Rhonda's business 15 years from the start.

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