Question

The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula \(x_{n+1}=b x_{n}\), where \(x_{n}\) is the population of houseflies at generation \(n\), and \(b\) is the average number of offspring per housefly who survive to the next generation. Assume a starting population \(x_{0}\). Find an expression for \(S_{n}=\sum_{i=0}^{n} x_{i}\) in terms of \(b\) and \(x_{0}\). What does it physically represent?

Short answer

\(S_n = x_0 \frac{1 - b^{n+1}}{1 - b}\) for total population over generations.

Step-by-step solution

Understanding the Population Model

The given recursive formula is \(x_{n+1} = b x_{n}\). This means that the population at the next generation \(n+1\) is the current population \(x_n\) multiplied by the average number of offspring \(b\). This is a geometric progression.

Find the Population Expression for Each Generation

Starting from \(x_0\), the subsequent populations can be expressed as follows:- \(x_1 = b x_0\)- \(x_2 = b x_1 = b^2 x_0\)- Continuing this pattern, \(x_n = b^n x_0\).

Setting Up the Series for Total Population

\(S_n\) is defined as the sum of populations from generation 0 to \(n\):\[ S_n = x_0 + x_1 + x_2 + \, \ldots \, + x_n = x_0 + b x_0 + b^2 x_0 + \, \ldots \, + b^n x_0 \]

Recognize the Series as a Geometric Series

The series \(x_0 + b x_0 + b^2 x_0 + \, \ldots \, + b^n x_0\) is a geometric series. It has a first term \(a = x_0\) and a common ratio \(r = b\).

Apply the Formula for the Sum of a Geometric Series

The sum of a geometric series with first term \(a = x_0\), common ratio \(r = b\), and \(n+1\) terms is given by:\[ S_n = x_0 \frac{1 - b^{n+1}}{1 - b} \] provided that \(b eq 1\).

Interpret the Physical Meaning of the Expression

The expression \(S_n = x_0 \frac{1 - b^{n+1}}{1 - b}\) represents the total number of houseflies over \(n\) generations. It accounts for the accumulation of houseflies across different generations.

Key concepts

Geometric Sequences

Geometric sequences are fascinating and play a crucial role in modeling population growth, like that of the housefly. In a geometric sequence, each term is derived by multiplying the previous term by a fixed, non-zero number called the common ratio. This is beautifully illustrated by the recursive formula given in the exercise, where housefly populations in each generation multiply by an average number of offspring, represented by the symbol \( b \).
Starting from an initial population \( x_0 \), the sequence progresses as follows:

• \( x_1 = b x_0 \)
• \( x_2 = b^2 x_0 \)
• \( x_3 = b^3 x_0 \)
• ... and so on

This pattern, where each new term is \( b \times \) the previous term, defines it as a geometric sequence. Understanding this concept helps us model and predict how populations change over generations.

Recursive Formulas

A recursive formula is a powerful tool that defines each term in a sequence using preceding terms. The exercise showcases the formula \( x_{n+1} = b x_n \), which presents a snippet of genetic and population cycles among organisms like houseflies.
Here's how it works:

• The term \( x_{n+1} \) represents the next generation's population.
• The term \( x_{n} \) is the current population.
• Multiplying \( x_{n} \) by \( b \) gives the next generation's population, considering \( b \) as the number of offspring that survive.

Given the initial term \( x_0 \), one can build the entire sequence, capturing the dynamics of population change. Recursive formulas are thus essential in creating realistic models of population growth and other sequential processes.

Sum of Geometric Series

The sum of a geometric series is an important concept in mathematics, especially in population modeling. The total population over multiple generations is an accumulation of each generation's contribution.
In this exercise, we consider the sum of populations from the start up to generation \( n \):

• \( S_n = x_0 + x_1 + x_2 + \ldots + x_n \)
• Which becomes \( S_n = x_0 + b x_0 + b^2 x_0 + \ldots + b^n x_0 \)

This is a geometric series where the first term \( x_0 \) and ratio \( b \) help calculate the cumulative sum. By the formula for the sum of a geometric series, the total is:\[ S_n = x_0 \frac{1 - b^{n+1}}{1 - b} \]provided \( b eq 1 \). This result reflects the total population over \( n \) generations and highlights the significance of both geometric sequences and recursive formulas in understanding cumulative growth.