Problem 33 - College Algebra

Question

Write the first eight terms of the sequence. $$a_{1}=2, a_{2}=10, a_{n}=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}}$$

Step-by-Step Solution

Answer

The first eight terms are 2, 10, 12, 2.8, 0.8, 2, 10, and 12.

Step 1: Understand the Sequence Formula

We are given the sequence formula: \(a_{n}=\frac{2(a_{n-1}+2)}{a_{n-2}}\). This tells us how to find the \(n\)-th term using the previous two terms of the sequence.

Step 2: Use Initial Values

The sequence starts with \(a_1 = 2\) and \(a_2 = 10\). These values are provided and needed to begin calculating subsequent terms in the sequence.

Step 3: Calculate the Third Term

To calculate \(a_3\), plug the given values into the sequence formula: \(a_3=\frac{2(a_2+2)}{a_1}=\frac{2(10+2)}{2}=12\).

Step 4: Calculate the Fourth Term

Next, find \(a_4\): \(a_4=\frac{2(a_3+2)}{a_2}=\frac{2(12+2)}{10}=2.8\).

Step 5: Calculate the Fifth Term

Now calculate \(a_5\) using the formula: \(a_5=\frac{2(a_4+2)}{a_3}=\frac{2(2.8+2)}{12}=0.8\).

Step 6: Calculate the Sixth Term

Find \(a_6\) as follows: \(a_6=\frac{2(a_5+2)}{a_4}=\frac{2(0.8+2)}{2.8}=2\).

Step 7: Calculate the Seventh Term

Now for \(a_7\): \(a_7=\frac{2(a_6+2)}{a_5}=\frac{2(2+2)}{0.8}=10\).

Step 8: Calculate the Eighth Term

Finally, calculate \(a_8\): \(a_8=\frac{2(a_7+2)}{a_6}=\frac{2(10+2)}{2}=12\).

Key Concepts

Sequence Formulas

Understanding sequence formulas is crucial when dealing with recursive sequences. A sequence formula provides a way to calculate terms in a sequence based on its defined relationship or rule. In the given exercise, the formula \(a_{n} = \frac{2(a_{n-1}+2)}{a_{n-2}}\) tells us how to generate any term based on the two preceding terms. This is a key feature of recursive sequences, as each term is typically derived from one or more previous terms using a specific formula.
Recursive formulas are essential for sequences where you cannot easily define an explicit formula for the \(n\)-th term. Instead, you rely on calculating each term step-by-step, starting from the given initial terms. This approach allows you to discover patterns and develop an understanding of the sequence's behavior over time.

Initial Values

Initial values are the foundation of recursive sequences. These are the starting terms from which all subsequent terms are calculated. For the sequence in this exercise, the initial values are \(a_1 = 2\) and \(a_2 = 10\).
With recursive sequences, knowing the initial values is absolutely necessary, as you need them to apply the sequence formula. Without these foundational values, you cannot compute any further terms in the sequence. Initial values provide the baselines or references from which the recursive process begins. Understanding their role helps in recognizing how sequences evolve and how each step builds upon the last.

Term Calculation

Calculating terms in a recursive sequence involves applying the sequence formula iteratively. In the given exercise, we progressively calculate each term using the formula \(a_{n} = \frac{2(a_{n-1}+2)}{a_{n-2}}\). Here's a breakdown of the method:

• Use known values: Begin with the initial values \(a_1 = 2\) and \(a_2 = 10\).
• Apply the formula: For each subsequent term, replace \(a_{n-1}\) and \(a_{n-2}\) with their known equivalents. This iterative process enables calculation of each term until the desired point in the sequence is reached.

Each application of the formula gives you the next term in the sequence, such as calculating \(a_3\), \(a_4\), and so on. Practicing this method not only reinforces an understanding of recursive techniques but also showcases the importance of accuracy in term substitution and computation.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that follow a specific rule or pattern. Sequences are essential in mathematics because they represent the progression of numerical patterns and can model real-world situations.
Types of sequences include arithmetic, geometric, and recursive, each having distinct characteristics:

• Arithmetic sequences have a constant difference between consecutive terms.
• Geometric sequences have a constant ratio between consecutive terms.
• Recursive sequences are determined by previous terms using a recursive formula, like in the exercise provided.

Recursive sequences, such as the one explored here, showcase the interplay between initial values and sequence formulas, emphasizing how a sequence evolves dynamically. Understanding the concept of mathematical sequences is essential for deeper explorations in mathematics and related fields.