Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, write the first eight terms of the sequence. \(a_{1}=2, \quad \mathrm{a}_{2}=10, \quad a_{n}=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}}\)

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 Problem

We are given a recursive sequence where \( a_1 = 2 \) and \( a_2 = 10 \). The recursive formula is \( a_n = \frac{2(a_{n-1} + 2)}{a_{n-2}} \). Our task is to find the first eight terms of the sequence.

Step 2: Calculate the Third Term

To find \( a_3 \), we use \( a_3 = \frac{2(a_2 + 2)}{a_1} = \frac{2(10 + 2)}{2} = 12 \). So, \( a_3 = 12 \).

Step 3: Calculate the Fourth Term

To find \( a_4 \), use \( a_4 = \frac{2(a_3 + 2)}{a_2} = \frac{2(12 + 2)}{10} = 2.8 \). So, \( a_4 = 2.8 \).

Step 4: Calculate the Fifth Term

For \( a_5 \), calculate \( a_5 = \frac{2(a_4 + 2)}{a_3} = \frac{2(2.8 + 2)}{12} = \frac{9.6}{12} = 0.8 \). Therefore, \( a_5 = 0.8 \).

Step 5: Calculate the Sixth Term

To find \( a_6 \), use \( a_6 = \frac{2(a_5 + 2)}{a_4} = \frac{2(0.8 + 2)}{2.8} = \frac{5.6}{2.8} = 2 \). So, \( a_6 = 2 \).

Step 6: Calculate the Seventh Term

Using \( a_6 \) and \( a_5 \), compute \( a_7 = \frac{2(a_6 + 2)}{a_5} = \frac{2(2 + 2)}{0.8} = 10 \). Therefore, \( a_7 = 10 \).

Step 7: Calculate the Eighth Term

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

Key Concepts

Sequence Terms

When dealing with sequences, especially recursive ones, the term 'sequence terms' becomes important. Sequence terms are the individual elements of a sequence. In this exercise, we aim to identify the first eight terms of the given recursive sequence. Each term is influenced by its predecessors, making the sequence dependently progressive. This means the values of earlier terms help determine the values of later terms. In our specific sequence:

• We know the first term is given: \( a_1 = 2 \).
• The second term is also provided: \( a_2 = 10 \).

With these initial conditions, you can begin to calculate each subsequent term using the provided recursive formula. Calculating further requires careful attention to ensure each step follows the formula's structure.

Recursive Formula

A recursive formula is a very special type of formula used to define the terms of a sequence with respect to the preceding terms. Instead of giving a direct formula for each term, it shows the relationship between them. In this exercise, the recursive formula is given by:\[ a_n = \frac{2(a_{n-1} + 2)}{a_{n-2}} \] For the third term \( a_3 \), you use both \( a_2 \) and \( a_1 \) to compute it. That's a unique feature of sequences defined recursively: they inherently require knowledge of initial terms. Each term is linked to the previous two terms. This demands precision, as any mistake in earlier terms could skew the entire sequence. Using recursive formulas thus requires meticulous calculations and careful adherence to the process.

Step-by-Step Solution

Breaking down the solution into steps helps to understand the computation of each term distinctly. Let's look at how each step contributes:

• **Step 1:** Clearly state the known values: \( a_1 = 2 \) and \( a_2 = 10 \). Start each computation with these values.
• **Step 2 to Step 7:** Calculate each term \( a_3 \) to \( a_8 \) one by one using the recursive formula. Ensure you substitute the correct terms in the place of \( a_{n-1} \) and \( a_{n-2} \). This means focusing on the relationships to obtain the next term step-by-step.

Following this methodical approach ensures that each term is verified before moving to the next. It simplifies the entire process into a series of manageable and understandable segments. This breakdown is essential for mastering sequences like this one, allowing for practice and accuracy.