Question

Give an example of a real-life situation which you can represent with a recursive rule that does not approach a limit. Write a recursive rule that represents the situation.

Short answer

A real-life situation that can be represented by a recursive rule without a limit is the annual population growth of a city which starts with a population of 10,000 and increases by 500 each year. The associated recursive rule is defined as follows: P(0) = 10000, P(n) = P(n-1) + 500 for n > 0. This situation does not approach a limit as the population continues to increase indefinitely.

Step-by-step solution

Identifying a Suitable Real-Life Situation

Identify a real-life situation that can be modeled using a recursive rule, and doesn't approach a limit. A suitable example is the population growth of a city.

Determine the Recursive Rule

Determine a recursive rule that models the situation. Suppose a city has a consistently increasing population. It starts out with a population of 10,000 and increases by 500 every year. This could be represented as follows: P(0) = 10000, which is the initial population, and P(n) = P(n-1) + 500, which is the recursive rule, where P(n) represents the population in year n.

Confirm the Absence of a Limit

The last step is to confirm that the defined recursive rule does not have a limit. Because the population increases by a constant amount each year, it doesn't approach a limit, but instead continues to increase indefinitely.

Key concepts

Population Growth

Population growth is a common example of a process that can be modeled using recursive formulas. Imagine a city where people are constantly moving in. This city starts with a population of 10,000 residents, and each year, 500 more people arrive. As years go by, new residents contribute to the overall growth.
This situation reflects growth that is constant over time, often due to factors like migration, birth rates, or economic opportunities. It's natural to want to describe such a dynamic using a formulaic approach. Leveraging a recursive rule, we aim to understand how the population changes year after year.

Recursive Rule

A recursive rule is a mathematical tool used to define a sequence where each term depends on one or more of the previous terms. In our example of city population growth, we use a recursive rule to describe how the population changes over time. Let's break it down:

• The initial population: \( P(0) = 10000 \)
• The population in any year: \( P(n) = P(n-1) + 500 \)

This rule tells us that the population \( P \) at any year \( n \) is the previous year's population plus 500. Recursive rules are useful because they simplify the modeling of processes where each step is dependent on the outcome of the previous step.
Whether you're tracking growth, decay or other sequential changes, recursive rules offer a consistent and logical framework for analysis.

Non-Converging Sequences

Non-converging sequences are another interesting concept when discussing recursive sequences. A sequence converges if it approaches a particular limit or value as time progresses. By contrast, a non-converging sequence keeps changing indefinitely and does not settle at any one point. Our city population growth example is a classic case of a non-converging sequence.
Since the population increases by 500 residents every year, it does not settle down to a constant number. Thus, there is no limit; the sequence of population values will keep getting larger indefinitely.
This feature can often be observed in systems or situations with consistent external influences or interventions, such as continuous population influx or constant economic investments.

Algebraic Modeling

Algebraic modeling allows us to use mathematical equations, including recursive rules, to represent real-life situations. This approach helps simplify complex systems by using mathematical symbols to describe changing scenarios, like population growth.
In the context of our city example:

• The initial value \( P(0) = 10000 \) represents the starting point of our model.
• The recursive rule \( P(n) = P(n-1) + 500 \) captures the ongoing growth process.

Using algebraic models like this makes it easier to predict future changes, analyze the dynamics at play, and make informed decisions. Whether you're dealing with economics, ecology, or demography, algebraic modeling provides a robust platform to explore and understand dynamic systems.