Problem 38 - Algebra 2 and Trigonometry

Question

In a small town, a census is taken at the beginning of each year. The census showed that there were \(5,000\) people living in the town at the beginning of 2001 and that the population decreased by 2\(\%\) each year for the next seven years. List the geometric sequence that gives the population of the town from 2001 to \(2008 .\) (A decrease of 2\(\%\) means that the population changed each year by a factor of \(0.98 .\) ) Write your answer to the nearest integer.

Step-by-Step Solution

Answer

5,000, 4,900, 4,802, 4,706, 4,612, 4,520, 4,430, 4,341

Step 1: Determine the First Term of the Sequence

The population at the beginning of 2001, which is the first term of the sequence, is given as 5,000.

Step 2: Identify the Common Ratio

The problem states that the population decreases by 2% each year. Therefore, the population changes by a factor of 0.98 each year. This is the common ratio in the geometric sequence.

Step 3: Calculate the Population for Each Year

To find the population for each year, multiply the previous year's population by the common ratio (0.98). Start with 5,000 for 2001: - 2001: 5,000 - 2002: \(5,000 \times 0.98 = 4,900\) - 2003: \(4,900 \times 0.98 \approx 4,802\) - 2004: \(4,802 \times 0.98 \approx 4,706\) - 2005: \(4,706 \times 0.98 \approx 4,612\) - 2006: \(4,612 \times 0.98 \approx 4,520\) - 2007: \(4,520 \times 0.98 \approx 4,430\) - 2008: \(4,430 \times 0.98 \approx 4,341\)

Step 4: List the Geometric Sequence

List the population figures starting from the year 2001 to 2008, rounding each to the nearest integer: \[5,000, 4,900, 4,802, 4,706, 4,612, 4,520, 4,430, 4,341\]

Key Concepts

Population Dynamics

Population dynamics refer to the changes in the size, composition, and distribution of populations over time. In this context, we're looking at a small town where the population is decreasing.
To analyze population dynamics, it's essential to consider factors that can lead to population changes, such as birth rates, death rates, immigration, and emigration.
In our example, we focus solely on a decrease due to a specific percentage reduction.

• **Initial Population**: In 2001, the town's population was 5,000.
• **Annual Change**: Each year, the population decreased in size due to an assumed 2% annual reduction.
• **Duration**: The population dynamics were considered over a span of seven years.

Observing population dynamics through this example gives insights into how a consistent percentage change can dramatically impact the population over a relatively short period.

Common Ratio

In the realm of a geometric sequence, the common ratio is a key concept. It denotes the constant factor by which the terms of the sequence are multiplied to obtain subsequent terms.
This factor is crucial when the sequence involves multiplicative growth or decay.In the provided example, the population decreases by 2% annually. To express this decrease in mathematical terms, it's necessary to understand that a 2% decrease is equivalent to retaining 98% of the previous value.
Thus, our common ratio here is:\[ r = 1 - 0.02 = 0.98 \]

• **0.98** is the scaling factor applied annually to represent the population decrease.
• This consistency allows us to predict the population for any given year within the period in question.

By applying this ratio, we can systematically calculate the remaining population each year, yielding a clear picture of the population dynamics over time.

Decrease by Percentage

When discussing population decreases by percentage, it's essential to grasp the math behind it, especially in sequences like the geometric ones described here.
A decrease by percentage means that each term is a certain percentage smaller than the previous one.
This type of decrease is quite common in real-world scenarios.

• **2% Decrease**: This means retaining 98% of the previous year's population, as mentioned in our common ratio.
• **Compounded Effect**: Over multiple years, this seemingly small decrease compounds, leading to a significantly lower population.

For our example, the formula \( P_{n} = P_{1} \cdot (0.98)^{n-1} \) is helpful to determine the population \( P_{n} \) for any year \( n \).
Here, \( P_{1} \) is the initial population of 5,000.Understanding how decreases by percentage work helps predict future population sizes and aids in planning and managing resources for communities over time.