Problem 32 - College Algebra

Question

For the following exercises, write the first eight terms of the sequence. $$ a_{1}=-1, a_{2}=5, a_{n}=a_{n-2}\left(3-a_{n-1}\right) $$

Step-by-Step Solution

Answer

The first eight terms are: -1, 5, 2, 5, -4, 35, 128, -4375.

Step 1: Understand the Sequence Definition

The sequence is defined recursively: we are given the first two terms: \(a_1 = -1\) and \(a_2 = 5\). Every subsequent term \(a_n\) for \(n \geq 3\) is calculated using the formula \(a_n = a_{n-2}(3 - a_{n-1})\).

Step 2: Calculate the Third Term

Use the formula to find \(a_3\):\[a_3 = a_1(3 - a_2) = -1(3 - 5) = -1(-2) = 2\].So, \(a_3 = 2\).

Step 3: Find the Fourth Term

Using the formula for \(a_4\):\[a_4 = a_2(3 - a_3) = 5(3 - 2) = 5 \times 1 = 5\].Thus, \(a_4 = 5\).

Step 4: Determine the Fifth Term

Calculate \(a_5\) using the prior terms:\[a_5 = a_3(3 - a_4) = 2(3 - 5) = 2(-2) = -4\].So, \(a_5 = -4\).

Step 5: Calculate the Sixth Term

Find \(a_6\):\[a_6 = a_4(3 - a_5) = 5(3 - (-4)) = 5 \times 7 = 35\].Therefore, \(a_6 = 35\).

Step 6: Determine the Seventh Term

Use the formula to find \(a_7\):\[a_7 = a_5(3 - a_6) = -4(3 - 35) = -4(-32) = 128\].Thus, \(a_7 = 128\).

Step 7: Find the Eighth Term

Calculate \(a_8\):\[a_8 = a_6(3 - a_7) = 35(3 - 128) = 35(-125) = -4375\].So, \(a_8 = -4375\).

Key Concepts

Sequence Calculation

A sequence is essentially an ordered list of numbers following a specific rule. In this exercise, we are tasked with finding the first eight terms of a sequence through sequence calculation. Each term is linked to other terms by a formula. This is why understanding the process of sequence calculation is crucial.

To start, we identify the initial terms provided: \(a_1 = -1\) and \(a_2 = 5\). These serve as the building blocks of our sequence. Each subsequent term, according to the recursive definition, relies on these.

Given the complexity that sequences can sometimes have, breaking down each step becomes important:

• First, write down the initial terms—this establishes a clear starting point.
• Use the recursive formula to find the next terms, noting how they rely on previously calculated terms.
• Calculate each step carefully to avoid cascading errors in later terms.

Thus, mastering sequence calculation involves understanding each part before moving to the next, ensuring accuracy and reliability in your results.

Recursion Formula

The recursion formula for a sequence is a way of defining each term with reference to its predecessors. This exercise makes use of such a formula. It can seem tricky at first, but it’s a vital tool in understanding sequence behavior.

The formula presented is \(a_n = a_{n-2} \times (3 - a_{n-1})\). This indicates that each term \(a_n\) is formed by multiplying two quantities: the term from two places back, \(a_{n-2}\), and the expression \(3 - a_{n-1}\), which involves the immediate preceding term.

Let’s break down the recursive process:

• Begin with the known initial values: \(a_1 = -1\) and \(a_2 = 5\).
• Apply the recursion formula to find \(a_3\). You use \(a_1\) and \(a_2\) as inputs.
• Systematically apply the same formula to find each subsequent term, using the relevant terms from previous calculations.

Recursion formulas allow for intricate relationships between terms, often leading to interesting patterns or behaviors in sequences. Being observant and methodical in applying these can lead to deeper insights into mathematical structures.

College Algebra

College Algebra is the foundation upon which a lot of more advanced mathematics is built. It involves understanding and manipulating symbols and expressions where identities, theorems, and functions come to play a crucial role.

In this exercise of recursive sequences, key elements of College Algebra are put to use—particularly in understanding how algebraic expressions and operations combine to result in sequences. Here, you observe:

• Algebraic manipulation: simplifying and working through operations involving integers and expressions.
• Recursive thinking: applying the formula repeatedly requires an understanding of how expressions transform.
• Symbolic representation: comprehending how symbols and terms interplay according to the rules of algebra.

Through recursive sequences, you see how a topic from College Algebra like function iteration becomes interactive, requiring you to apply theoretical knowledge in hands-on calculations. This not only strengthens algebra skills but also enhances your problem-solving capabilities.