Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

The sequence defined recursively by \(x_{k+1}=x_{k} /\left(1+x_{k}\right)\) occurs in genetics in the study of the elimination of a deficient gene from a population. Show that the seauence whose \(n\)th term is \(1 / x\). is arithmetic.

Step-by-Step Solution

Answer

The sequence \( \frac{1}{x_k} \) is arithmetic with a common difference of 1.

Step 1: Understanding the Sequence

The sequence is defined recursively with a formula for the next term based on the current term: \( x_{k+1} = \frac{x_k}{1 + x_k}\). We need to show that the sequence formed by \( \frac{1}{x_k} \) is arithmetic.

Step 2: Define the Inverse Sequence

Let \( y_k = \frac{1}{x_k} \) so that the sequence becomes \( y_k \) instead of \( x_k \). Our goal is to express \( y_{k+1} \) in terms of \( y_k \) and check for arithmetic properties.

Step 3: Derive the Relation for the Inverse

By definition, \( y_{k} = \frac{1}{x_k} \). We need to find \( x_{k+1} \) in terms of \( y_k \):\[ x_{k+1} = \frac{x_k}{1 + x_k} \rightarrow \frac{1}{y_{k+1}} = \frac{1/y_k}{1 + 1/y_k} \]

Step 4: Simplify the Expression

Simplify the expression for \( y_{k+1} \):\[ \frac{1}{y_{k+1}} = \frac{1}{y_k} \cdot \frac{y_k}{y_k + 1} = \frac{1}{y_k + 1} \]Thus, \( y_{k+1} = y_k + 1 \).

Step 5: Identify as an Arithmetic Sequence

Notice that the recursive formula \( y_{k+1} = y_k + 1 \) describes an arithmetic sequence, where the common difference is 1.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a series of numbers where each term increases or decreases by a constant difference, known as the common difference. Imagine a sequence of numbers: 2, 4, 6, 8. Each number is formed by adding a constant, in this case, 2, to its previous term. That's what makes it arithmetic.

This property makes arithmetic sequences straightforward to understand and predict. Once you know the first term and the common difference, you can determine any term in the sequence. The equation for finding the nth term of an arithmetic sequence is:

• \( a_n = a_1 + (n-1) \cdot d \)

where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference. In our solution, the inverse sequence was shown to be arithmetic because each term \( y_{k+1} \) is simply the previous term \( y_k \) plus 1, leading to a common difference of 1.

Inverse Sequence

Understanding inverse sequences adds depth to solving complex problems, such as determining arithmetic properties within recursive sequences. An inverse sequence refers to taking the reciprocal of each term in a given sequence. For example, converting a sequence \( x_k \) into an inverse sequence \( y_k = \frac{1}{x_k} \).

In our exercise, by establishing \( y_k = \frac{1}{x_k} \), we could then focus on the new sequence of terms formed by these reciprocals. We explored how this transformation impacts the nature of the sequence. Simplifying this resulted in a relationship \( y_{k+1} = y_k + 1 \).

This reveals that the original challenge of working with a complex recursive relation translates into identifying an arithmetic sequence for the inverse, as each new term is formed by adding a constant to the previous term.

Genetics in Population Studies

Sequences and their properties play a crucial role in genetics, particularly when studying how traits pass through generations or how certain genes may be eliminated through selective breeding. Recursive sequences are often used to model such phenomena as they can easily represent numerous generations.

In our specific exercise, the recursive sequence \( x_{k+1} = \frac{x_k}{1 + x_k} \) models the elimination of a deficient gene from a population. By analyzing the inverse sequence, which formed an arithmetic progression, it becomes easier to grasp how the frequency of this gene changes over time.

Understanding this makes it simpler to predict how long it might take for a gene to diminish significantly in a population. These recursive models are indispensable for geneticists to forecast and understand evolutionary changes or outcomes of particular breeding strategies.