Problem 33 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$a_{0}=2, a_{1}=3, \text { and } a_{n}=\left(a_{n-1}\right)\left(\frac{1}{2} a_{n-2}\right) \quad \text { for } n \geq 2$$

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are \(a_0 = 2\), \(a_1 = 3\), \(a_2 = 3\), \(a_3 = \frac{9}{2}\), and \(a_4 = \frac{27}{4}\).

Step 1: Write Down the Given Information

We have been given the first two terms of the sequence and the recursive relationship: $$a_{0} = 2, a_{1} = 3, a_{n} = a_{n-1}(\frac{1}{2}a_{n-2}) \quad \text{ for } n \geq 2$$

Step 2: Find the Third Term (n=2)

Apply the recursive relationship to find \(a_2\): $$a_2 = a_{1}\left(\frac{1}{2}a_{0}\right) = 3 \left(\frac{1}{2} \cdot 2\right) = 3$$

Step 3: Find the Fourth Term (n=3)

Apply the recursive relationship again to find \(a_3\): $$a_3 = a_{2}\left(\frac{1}{2}a_{1}\right) = 3 \left(\frac{1}{2} \cdot 3\right) = \frac{9}{2}$$

Step 4: Find the Fifth Term (n=4)

Apply the recursive relationship once more to find \(a_4\): $$a_4 = a_{3}\left(\frac{1}{2}a_{2}\right) = \frac{9}{2} \left(\frac{1}{2} \cdot 3\right) = \frac{27}{4}$$

Step 5: Write Down the First Five Terms

We have now calculated the first five terms of the sequence: $$a_0 = 2,\, a_1 = 3,\, a_2 = 3,\, a_3 = \frac{9}{2},\, a_4 = \frac{27}{4}$$

Key Concepts

Recurrence Relation

A recurrence relation is a mathematical expression that defines each term of a sequence using the preceding terms. This type of relation is crucial because it allows us to generate sequences by building upon previous values, enabling complex patterns to emerge from simple rules. The given exercise shows a classic example of a recurrence relation, where each term from the third onwards is determined by the formula \(a_n = a_{n-1}(\frac{1}{2}a_{n-2})\). Here, the sequence starts with known initial values of \(a_0 = 2\) and \(a_1 = 3\), establishing a base from which further terms can be calculated. Recurrence relations are quite common in computer algorithms and can represent real-world processes like population growth or financial models.

Sequence Terms

Sequence terms are the individual elements in a sequence, each identified by its position, or index, in the sequence. In the context of a recursively defined sequence, the initial terms are explicitly given—as in our case \(a_0 = 2\) and \(a_1 = 3\). For these terms, no calculations are needed beyond knowing their values.
Understanding the initial terms is essential because they serve as the foundation for calculating subsequent terms using the recurrence relation.

• The first term \(a_0\) provides the starting point.
• The second term \(a_1\) helps to initiate the recurring pattern.
• The further terms like \(a_2, a_3, a_4\), etc., are derived based on these initial values using the relation provided.

Each term's role is significant as each acts as a stepping stone for the progression of the sequence.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers characterized by specific rules that dictate how each term is correlated to the others. In particular, recursively defined sequences utilize recurrence relations to specify these rules. The sequence in our exercise is a perfect illustration where after establishing the first two terms, each subsequent term is calculated using preceding ones. This structure is fundamental because:

• It helps predict future terms once the pattern is clear.
• Sequences such as arithmetic or geometric sequences are built on related principles, showing how diverse sequence structures can be interconnected.
• Manipulating sequences can lead to complex functions and models beneficial for scientific analysis.

Understanding sequences and their construction like our recursive one is key in areas ranging from mathematics to computer science.

Step-by-Step Solution

Breaking down solutions into steps is invaluable for tackling complex mathematical problems. In this exercise, we solved for the first five terms of a sequence using step-by-step instructions. Here’s how the process unfolded:
- **Step 1:** Identified initial terms \(a_0 = 2\) and \(a_1 = 3\).
- **Step 2:** Calculated \(a_2\) by substituting into the recurrence relation \(a_2 = a_1(\frac{1}{2}a_0) = 3\).
- **Step 3:** Derived \(a_3\) likewise using \(a_3 = a_2(\frac{1}{2}a_1) = \frac{9}{2}\).
- **Step 4:** Concluded with \(a_4\) where \(a_4 = a_3(\frac{1}{2}a_2) = \frac{27}{4}\).

This meticulous breakdown ensures each part of the sequence is built correctly and understandably, reinforcing the power of organized, detailed problem-solving.