Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Find the first four terms of each sequence. $$a_{1}=2, a_{2}=1, a_{n}=2 a_{n-1}^{2}+a_{n-2}, \text { for } n>2$$

Step-by-Step Solution

Answer

The first four terms are 2, 1, 4, and 33.

Step 1: Understand the Sequence Formula

Read the given formula for the sequence: \(a_1 = 2\), \(a_2 = 1\), and \(a_n = 2a_{n-1}^2 + a_{n-2}\) for \(n > 2\). This means that each term after the second is calculated using the formula and the two preceding terms.

Step 2: Calculate the First Term

The first term of the sequence is given as \(a_1 = 2\).

Step 3: Calculate the Second Term

The second term of the sequence is given as \(a_2 = 1\).

Step 4: Calculate the Third Term

Use the formula to find \(a_3 = 2a_2^2 + a_1\). Substituting the known values gives \(a_3 = 2(1)^2 + 2 = 4\).

Step 5: Calculate the Fourth Term

Use the formula to find \(a_4 = 2a_3^2 + a_2\). Substitute the known values: \(a_4 = 2(4)^2 + 1 = 2(16) + 1 = 32 + 1 = 33\).

Key Concepts

Recursion

Recursion is a simple yet powerful concept in sequences and mathematics. It involves defining the elements of a sequence through their predecessors using a specific rule or formula. In the given exercise, the sequence is defined recursively because each term is calculated based on the values of the previous terms. This type of definition is prevalent in sequences, particularly when the pattern is complex and cannot be easily expressed with a single formula.
When studying recursion, it's helpful to start with a base case, which in this exercise is provided as \(a_1 = 2\) and \(a_2 = 1\). These initial values kick-start the recursive process. Then, for \(n > 2\), the recursive formula \(a_n = 2a_{n-1}^2 + a_{n-2}\) guides us in finding each subsequent term by utilizing the previously calculated terms. Understanding recursion is crucial because it builds the foundation for understanding how sequences progress.

Sequence Formula

The sequence formula is the mathematical expression used to calculate the terms of a sequence. In the exercise, the sequence formula given is \(a_n = 2a_{n-1}^2 + a_{n-2}\) for \(n > 2\). Such a formula is essential as it establishes the relationship between any term and its predecessors.
The provided formula uses the terms \(a_{n-1}\) and \(a_{n-2}\) to find the \(n\)-th term. This demonstrates how each term in a sequence can depend on multiple previous terms, which is common in complex sequences.

• Begin by identifying the initial terms, such as \(a_1 = 2\) and \(a_2 = 1\).
• Use these to apply the formula iteratively to discover additional terms.

Understanding and correctly applying the sequence formula is key to solving problems related to sequence determination.

Term Calculation

Term calculation in sequences involves using a sequence formula to determine specific terms. For our sequence, once you have the initial terms \(a_1\) and \(a_2\), you can calculate the subsequent terms using the recursive formula.
To calculate \(a_3\), you should follow these steps:

• Interchange \(n\) in the formula with the term number you're calculating, so for \(a_3\), plug in \(n = 3\).
• Use known values: \(a_3 = 2a_2^2 + a_1\).
• Since \(a_2 = 1\) and \(a_1 = 2\), plugging in these values gives \(a_3 = 2(1)^2 + 2 = 4\).

Similarly, calculating the fourth term requires using the previously calculated term \(a_3\):

• Use the formula \(a_4 = 2a_3^2 + a_2\).
• Substitute \(a_3 = 4\) and \(a_2 = 1\), resulting in \(a_4 = 2(4)^2 + 1 = 33\).

Each calculated term relies directly on its predecessors, showcasing the power of recursive sequences.

Mathematical Induction

Mathematical induction is a technique often used to prove statements or formulas related to sequences. However, in this specific exercise, we're more focused on its theoretical application rather than a direct proof. An important point about mathematical induction is that it establishes the truth of an infinite sequence of propositions.
Here's a general approach to understand its principles:

• Base Case: Verify the statement for the initial terms, \(a_1\) and \(a_2\). Often, this base verification is a simple computation.
• Inductive Step: Assume the formula holds for a term \(a_k\). Then, prove it true for \(a_{k+1}\). Essentially, if it holds for one term, it holds for the next.
• Once both steps are applied successfully, you conclude the formula or statement is valid for all subsequent terms.

Although induction isn't explicitly calculated in this exercise, understanding it helps recognize the reliability and correctness of recursive formulas in sequences.