Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

\( a_{1}=3, \quad a_{k+1}=1 / a_{k}\)

Step-by-Step Solution

Answer

The sequence alternates between 3 and \(\frac{1}{3}\) with a period of 2.

Step 1: Understanding the Sequence

The given sequence is defined recursively. The first term, \(a_1\), is 3. The rule for consecutive terms is given by \(a_{k+1} = \frac{1}{a_k}\). This means each term is the reciprocal of the previous term.

Step 2: Calculate the First Few Terms

Start with the first term: - \(a_1 = 3\)Calculate the next terms using the rule:- \(a_2 = \frac{1}{a_1} = \frac{1}{3}\)- \(a_3 = \frac{1}{a_2} = 3\)- \(a_4 = \frac{1}{a_3} = \frac{1}{3}\)

Step 3: Identify the Pattern

Notice that the terms alternate between 3 and \(\frac{1}{3}\). This suggests the sequence is periodic with a period of 2 terms.

Key Concepts

Recurrence Relations

Recurrence relations are fundamental when dealing with sequences. They describe each term in a sequence by relating it to one or more of its predecessors. In the given exercise, the recurrence relation is shown as \(a_{k+1} = \frac{1}{a_k}\). This means the next term is always the reciprocal of the current term.

Recurrence relations can help in understanding the complex behavior of sequences without having to directly compute many terms. They allow for the establishment of patterns or rules that are consistent throughout the sequence.

• Identify the base case (first term)
• Determine the relation that gives subsequent terms
• Look for patterns or attributes such as periodicity

Understanding these mathematical expressions makes it easier to solve problems about sequences and can offer insights into different mathematical fields.

Periodic Sequences

Periodic sequences are sequences that repeat themselves after a certain number of terms. In the context of the given problem, identifying that a sequence is periodic can simplify analysis as it reduces infinite complexity to a repeating cycle.

With the exercise given, the sequence alternates between 3 and \(\frac{1}{3}\). This alternation pattern indicates the sequence is periodic with a period of 2 terms. A sequence is defined as periodic if there exists some integer \(P\) such that \(a_{n+P} = a_n\) for all integers \(n\).

To identify a periodic sequence:

• Calculate the initial terms to find repeating patterns
• Verify that the patterns are consistent across the terms
• Determine the period (least number of terms before repetition)

Acknowledging periodicity helps understand the long-term behavior of a sequence without complex calculations.

Sequence Convergence

Sequence convergence refers to the idea that the terms in a sequence approach a specific value, known as the limit, as the number of terms increases. In simpler terms, as you go further and further along the sequence, the terms get closer to a specific number.

However, for the sequence given in the exercise, which alternates between 3 and \(\frac{1}{3}\), convergence does not occur. The sequence does not get closer to any single number because it is periodic. Periodic sequences usually do not converge to a single value because they keep repeating a set of values rather than stabilizing to one specific term.

• Recognize if the sequence gets steadily closer to a number
• Identify if the sequence oscillates or diverges
• Determine if the terms eventually stabilize

Understanding sequence convergence is important because it allows us to predict the behavior of sequences in the long term. But always remember, not all sequences converge.