Exploring Real-World Scenarios- Which Situations Can Be Modeled with Geometric Sequences-

Which situation could be modeled using a geometric sequence?

Geometric sequences are a fundamental concept in mathematics, characterized by a constant ratio between consecutive terms. These sequences are widely used to model various real-world situations where the rate of change is proportional to the current value. One such situation is the growth of a population over time, which can be effectively represented using a geometric sequence.

In the field of biology, population growth is a classic example of a situation that can be modeled using a geometric sequence. When a population grows at a constant rate, each generation is larger than the previous one by a fixed multiple. This exponential growth can be represented by a geometric sequence, where the first term is the initial population size, and the common ratio is the growth rate.

For instance, consider a population of bacteria that doubles in size every hour. If we start with an initial population of 100 bacteria, the population at each subsequent hour can be calculated using the geometric sequence formula:

P(n) = P(0) r^n

where P(n) is the population at time n, P(0) is the initial population, r is the common ratio (in this case, 2), and n is the number of time intervals.

At the end of the first hour, the population will be 100 2 = 200 bacteria. At the end of the second hour, it will be 200 2 = 400 bacteria, and so on. This pattern continues, illustrating the exponential growth of the population over time.

Another example of a situation that can be modeled using a geometric sequence is the depreciation of an asset. When an asset depreciates at a constant rate, the value of the asset decreases by a fixed percentage each year. This can be represented by a geometric sequence, where the first term is the initial value of the asset, and the common ratio is 1 minus the depreciation rate.

For instance, consider a car that depreciates by 20% each year. If the car’s initial value is $30,000, the value of the car at the end of each year can be calculated using the geometric sequence formula:

V(n) = V(0) (1 – r)^n

where V(n) is the value of the car at time n, V(0) is the initial value, r is the common ratio (in this case, 0.2), and n is the number of years.

At the end of the first year, the car’s value will be $30,000 (1 – 0.2) = $24,000. At the end of the second year, it will be $24,000 (1 – 0.2) = $19,200, and so on. This pattern continues, illustrating the depreciation of the car’s value over time.

In conclusion, geometric sequences are a powerful tool for modeling situations where the rate of change is proportional to the current value. Examples such as population growth and asset depreciation demonstrate the versatility of geometric sequences in representing real-world phenomena. By understanding and applying these sequences, we can gain valuable insights into various aspects of our daily lives.