A nursery grows aloe plants for sale. Every month, the nursery takes a cutting from each of the aloe plants and plants it in a new pot. Each cutting creates one new aloe plant. At the end of the month, the nursery sells 100 plants to a local landscaper. Let an be the number of aloe plants after n months. Select the recurrence relation that describes the sequence {an}.

Question

BlackTom AI · Accepted Answer

Each plant produces one cutting so the total count doubles before any sales; then 100 plants are sold at the end of the month. This yields a straightforward recurrence.

1. Relevant knowledge points and concepts
- Recurrence relations describe how a term a_n depends on earlier terms.
- The process: from a_{n-1} plants you get a_{n-1} new plants (one cutting per plant), so the population immediately becomes 2·a_{n-1}; then 100 plants are removed (sold).

2. Step-by-step reasoning
- Start with a_{n-1}, the number of plants at the start of month n.
- Cuttings: each of those produces one new plant, adding a_{n-1} more.
- Count after cuttings (but before sales): a_{n-1} + a_{n-1} = 2·a_{n-1}.
- Sales at the end of the month remove 100 plants.
- Therefore a_n = (2·a_{n-1}) − 100.

3. Intermediate calculation example
- Example: if a_0 = 500, after cuttings the count is 2·500 = 1000. After selling 100, a_1 = 1000 − 100 = 900. This matches the formula a_1 = 2·a_0 − 100 = 900.

4. Why the correct answer is correct
- The order (doubling then subtracting 100) matches the described timeline: cuttings occur during the month (creating new plants), and sales happen at the end. Algebraically and conceptually this is exactly a_n = 2·a_{n-1} − 100.

5. Why the other choices are incorrect
- a_n = 2(a_{n-1} − 100) expands to 2·a_{n-1} − 200, which would mean 100 plants are removed before cuttings (or that 200 are effectively removed after doubling); this contradicts the stated order of events.
- a_n = a_{n-1} + a_{n-2} − 100 and a_n = a_{n-1} + a_{n-2} − 200 imply dependence on two previous months (Fibonacci-like reproduction), which is not described by the problem.

6. Conclusion
The correct recurrence is an=2⋅an−1−100, which matches the provided answer.

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