Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Find the first four terms of each sequence. $$a_{1}=2, a_{2}=3, a_{n}=a_{n-1} \cdot a_{n-2} \text { for } n>2$$

Step-by-Step Solution

Answer

The first four terms are 2, 3, 6, and 18.

Step 1: Identify Known Terms

The first term of the sequence is given as \(a_1 = 2\) and the second term is \(a_2 = 3\). These are the initial conditions that allow us to calculate the subsequent terms.

Step 2: Calculate Third Term

Using the formula \(a_n = a_{n-1} \cdot a_{n-2}\), we calculate \(a_3\): \(a_3 = a_2 \cdot a_1 = 3 \cdot 2 = 6\).

Step 3: Calculate Fourth Term

Using the same formula \(a_n = a_{n-1} \cdot a_{n-2}\), calculate \(a_4\):\(a_4 = a_3 \cdot a_2 = 6 \cdot 3 = 18\).

Key Concepts

Recursive Formula

A recursive formula provides a convenient way to define a sequence where each term is computed from the previous ones. For many sequences, it's not practical to find every single term manually. Instead, a recursive formula comes to the rescue by establishing a pattern based on preceding terms. In our example, the sequence is given by the formula: \[ a_n = a_{n-1} \cdot a_{n-2}, \text{ for } n>2 \] This means that for any term after the second one, you multiply the two terms that come before it. This algebraic relationship quickly lets us find each subsequent term by using already known ones. Such formulas make sequences both predictable and easily calculable, perfect tools for understanding how sequences progress and create patterns.

Initial Conditions

Initial conditions are the building blocks that kick-start a recursive sequence. They set the framework upon which all other terms are calculated. In our example, the initial conditions are given as:

• \( a_1 = 2 \)
• \( a_2 = 3 \)

By defining these starting values, the rest of the sequence can be unfolded systematically using the recursive formula. Without them, we would have no foundation to construct further terms. Initial conditions encapsulate the sequence's distinct qualities and ensure the correct execution of the pattern.

Product of Terms

In the given sequence, each new term is the product of its two preceding terms. This concept highlights how multiplying terms together creates the next term in the series. For instance, using the recursive formula for our specific sequence:

• To find \( a_3 \), you calculate \( 3 \times 2 = 6 \).
• For \( a_4 \), multiply \( 6 \times 3 = 18 \).

Each term grows based on these products, illustrating how values can change dramatically or remain stable depending on the nature of multiplication at play. This principle of using earlier values to compute a next one emphasizes the multiplicative aspect of recursive relations and how they generate surprising results.