Problem 81 - A Graphical Approach to College Algebra

Question

(Modeling) Fruit Fly Populations A population of fruit flies is growing in such a way that each generation is 1.5 times as large as the last generation. If there were 100 flies in the first generation, write a formula for a geometric sequence that gives the population of the \(n\) th generation. Find the population of the fourth generation.

Step-by-Step Solution

Answer

The fourth generation has 338 fruit flies.

Step 1: Recognizing the Type of Sequence

The problem describes a situation where each generation is 1.5 times the previous one. This is a classic example of a geometric sequence, where each term is found by multiplying the previous term by a constant factor.

Step 2: Identify the First Term and Common Ratio

In a geometric sequence, we start with a first term and multiply it by a common ratio to get the next terms. Here, the first term \( a_1 \) is 100 flies, and the common ratio \( r \) is 1.5.

Step 3: Writing the Formula for the Geometric Sequence

The general form for a geometric sequence is \( a_n = a_1 \, r^{(n-1)} \). For this specific problem, the formula for the \( n \)th generation is: \[ a_n = 100 \, (1.5)^{(n-1)} \]

Step 4: Calculate the Population of the Fourth Generation

Substitute \( n = 4 \) into the formula to find the population of the fourth generation. \[ a_4 = 100 \, (1.5)^{(4-1)} = 100 \, (1.5)^3 \]

Step 5: Simplifying the Calculation

Calculate \( (1.5)^3 \): \( 1.5 \times 1.5 = 2.25 \), and \( 2.25 \times 1.5 = 3.375 \). Therefore, \( (1.5)^3 = 3.375 \).

Step 6: Final Computation

Now, find \( a_4 \) by multiplying: \[ a_4 = 100 \, \times 3.375 = 337.5 \]

Step 7: Round to Whole Number

Since flies can't be in fractions, rounding 337.5 gives 338 flies.

Key Concepts

Population Growth

Population growth in mathematical modeling often involves understanding how a population of organisms increases over time. A typical example is fruit fly populations, where we observe changes from one generation to the next.
In this scenario, each generation of fruit flies grows by a fixed percentage, specifically by a factor of 1.5 from the previous generation. This means the population doesn't just add a consistent number every time. Instead, it multiplies, which is characteristic of exponential growth.
For instance, if you start with 100 flies, each new generation will have 1.5 times the number of flies as the prior generation. This kind of growth can be represented using a geometric sequence, which is a crucial tool in understanding such biological phenomena.

Common Ratio

In a geometric sequence, understanding the common ratio is essential, as it directly influences how each term progresses from one to the next.
The common ratio is the factor by which we multiply a term to move to the following term in the sequence. For the fruit fly example, this common ratio is 1.5. This means each new generation has 1.5 times as many fruit flies as the previous one.

• Common Ratio Formula: In a sequence where the first term is known, the general form is \( a_n = a_1 \cdot r^{(n-1)} \).
• Significance: The common ratio can tell us whether the sequence is increasing, decreasing, or constant over time.

Understanding the common ratio allows us to predict future population sizes easily and interpret the growth trends reliably.

Formula derivation

Deriving the formula for a geometric sequence involves recognizing the consistent multiplication by the common ratio. Let’s break down each step for clarity using the fruit fly problem as an example.
First, identify the initial values:

• Initial population \( a_1 = 100 \) flies.
• Common ratio \( r = 1.5 \).

Using these values, insert them into the general formula for a geometric sequence: \[ a_n = a_1 \cdot r^{(n-1)} \] Thus, the specific formula for the fruit fly population is:\[ a_n = 100 \cdot (1.5)^{(n-1)} \]
Now, to find the population of the fourth generation, substitute \( n = 4 \) into the formula:
\[ a_4 = 100 \cdot (1.5)^{3} \]
By calculating the powers and multiplying by 100, we obtain: \( a_4 = 100 \cdot 3.375 = 337.5 \). Rounding yields 338 flies.
Showing each of these detailed steps makes the derivation process less daunting and more approachable for students.