Question

Plot the first \(N\) terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. $$ \text { [7] } a_{1}=1, a_{2}=2, \text { and for } n \geq 3, a_{n}=\sqrt{a_{n-1} a_{n-2}} ; N=30 $$

Short answer

The sequence converges, as evidenced by the terms stabilizing around a specific value.

Step-by-step solution

Understanding the Sequence

The given sequence is defined by the first two terms: \( a_1 = 1 \) and \( a_2 = 2 \). For terms where \( n \geq 3 \), a recursive definition is provided: \( a_n = \sqrt{a_{n-1} a_{n-2}} \). We need to compute the first 30 terms of this sequence.

Calculating Initial Terms

We begin by listing the terms we already have: \( a_1 = 1 \) and \( a_2 = 2 \). Next, we need to calculate \( a_3 \) using the recursive definition: \( a_3 = \sqrt{a_2 a_1} = \sqrt{2 \times 1} = \sqrt{2} \).

Calculating Subsequent Terms

Continue calculating terms using the recursion. For \( a_4 \), use \( a_4 = \sqrt{a_3 a_2} = \sqrt{\sqrt{2} \times 2} = \sqrt{2} \). Repeat this process for each subsequent term up to \( a_{30} \). Each new term should be calculated using the two previous terms.

Generating the Plot

Plot the computed terms on a graph with the term number \( n \) on the x-axis and the value of \( a_n \) on the y-axis. This will help visualize how the sequence behaves as \( n \) grows from 1 to 30.

Analyzing the Graph

Examine the graph to determine whether the sequence seems to converge or diverge. Check if the terms approach a specific value or if they continue to change without settling to a particular limit. A visual inspection should reveal if the sequence converges.

Conclusion on Convergence or Divergence

Based on the graph, if the terms appear to stabilize around a certain value, the sequence converges. If the values of \( a_n \) keep oscillating or moving away indefinitely, the sequence diverges.

Key concepts

Graphical Analysis

Graphical analysis is a powerful tool when you want to understand the behavior of a sequence. It involves plotting the terms of a sequence on a graph to visually inspect how the sequence evolves with each new term. On a graph, the x-axis typically represents the term number, and the y-axis shows the value of each term. This visual representation can make it easier to see patterns or trends that might not be obvious through numerical data alone.

• If the terms of the sequence appear to approach a fixed value, the graph indicates convergence.
• If the terms continuously veer away without settling, it shows divergence.

By plotting the first 30 terms of the given sequence, you can observe whether the sequence seems to settle into a pattern, like steady convergence towards a specific value, or exhibits erratic behavior that suggests divergence. What you are looking for is whether the points on the graph start clustering around a line or a point, indicating a limit.

Recursive Sequences

Recursive sequences define a sequence's terms based on previous terms. In the given exercise, the sequence is defined recursively for liabilities from the third term onwards. A recursive formula relies on a base case—here, the first two terms, known as initial conditions. The sequence continues by applying the same rule repeatedly:
For the given sequence:

• Initial terms, \(a_1 = 1\) and \(a_2 = 2\), are explicitly given.
• For any subsequent term \(a_n\), where \(n \geq 3\), the sequence is defined as \(a_n = \sqrt{a_{n-1} a_{n-2}}\).

This recursive nature allows the sequence to be computed step-by-step, where each term is derived from its predecessors. It is this characteristic that makes recursive sequences both interesting and powerful in mathematical modeling.

Term Calculation

To analyze a recursive sequence, calculating its terms is essential. This ensures you have the data needed for graphical analysis and understanding the overall sequence behavior. Here is how you proceed with term calculation in the given sequence:

• Start with known initial terms: \(a_1 = 1\) and \(a_2 = 2\).
• Calculate \(a_3\) using the recursive formula: \(a_3 = \sqrt{a_2 \times a_1} = \sqrt{2}\).
• Continually calculate each term up to the desired \(a_{30}\), using \(a_n = \sqrt{a_{n-1} \times a_{n-2}}\).

For each term, substitute the values of the previous two terms, perform the square root operation, and proceed to the next term. This consistent methodical approach not only assists in plot generation but also deepens one's understanding of recursive behavior and convergence in sequences.