Problem 32 - College Algebra

Question

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 Given Information

We are given a sequence that starts with initial terms \(a_1 = -1\) and \(a_2 = 5\). The recursive formula for the sequence is given as \(a_n = a_{n-2}(3 - a_{n-1})\). This formula will help us find each subsequent term based on the two preceding terms.

Step 2: Calculate the Third Term

Using the recursive formula \(a_3 = a_1(3 - a_2)\). Substitute the given values: \(a_3 = (-1)(3 - 5) = (-1)(-2) = 2\). Hence, the third term is \(a_3 = 2\).

Step 3: Calculate the Fourth Term

Use the formula with \(n=4\): \(a_4 = a_2(3 - a_3)\). Substituting the known values gives us \(a_4 = 5(3 - 2) = 5 \). So, \(a_4 = 5\).

Step 4: Calculate the Fifth Term

Next, find \(a_5\) using the formula \(a_5 = a_3(3 - a_4)\). Substituting the values: \(a_5 = 2(3 - 5) = 2(-2) = -4\). Therefore, \(a_5 = -4\).

Step 5: Calculate the Sixth Term

For \(n=6\), compute \(a_6 = a_4(3 - a_5)\). Substituting \(a_4 = 5\) and \(a_5 = -4\), we find \(a_6 = 5(3 - (-4)) = 5(3 + 4) = 35\). So, \(a_6 = 35\).

Step 6: Calculate the Seventh Term

Using the formula for \(n=7\), we have \(a_7 = a_5(3 - a_6)\). Substitute \(a_5 = -4\) and \(a_6 = 35\), giving \(a_7 = -4(3 - 35) = -4(-32) = 128\). Thus, \(a_7 = 128\).

Step 7: Calculate the Eighth Term

For \(a_8\), substitute into the formula: \(a_8 = a_6(3 - a_7)\). Use the previous results: \(a_8 = 35(3 - 128) = 35(-125) = -4375\). Hence, \(a_8 = -4375\).

Step 8: List the First Eight Terms

After calculating each term individually, the first eight terms of the sequence are: \(a_1 = -1\), \(a_2 = 5\), \(a_3 = 2\), \(a_4 = 5\), \(a_5 = -4\), \(a_6 = 35\), \(a_7 = 128\), \(a_8 = -4375\).

Key Concepts

Sequence Problems

Sequence problems are a fascinating area of study in mathematics. A sequence is a set of numbers arranged in a specific order, following an identifiable pattern. Problems with sequences often involve finding the rule that defines the sequence and predicting future terms. Students are typically given either an explicit formula or a recursive formula to generate the terms.

• Explicit formula: This provides a direct method to calculate any term in the sequence without relying on previous terms.
• Recursive formula: This calculates each term based on one or more preceding terms.

In the original exercise, we were given a recursive sequence. The challenge here was not just in simple calculation, but in understanding how each term is dependent on its predecessors. Recognizing the pattern laid out by the recursive rule is key to solving these problems.

Recursive Formulas

Recursive formulas are an intriguing way to define sequences because each term relies on previous ones. The key aspect is the initial terms given to start the sequence and the recursive relationship that connects them. In our exercise, the sequence begins with two terms: \(a_1 = -1\) and \(a_2 = 5\). The recursive formula is \(a_n = a_{n-2}(3 - a_{n-1})\). The goal is to use this formula to find subsequent terms.

• This means to find \(a_3\), you would use \(a_1\) and \(a_2\).
• For \(a_4\), you use \(a_2\) and \(a_3\).

Recursive sequences, though sometimes tricky to handle, provide a powerful way to generate complex patterns and behaviors from simple rules. Understanding this foundational concept boosts your ability to handle various sequence problems, laying groundwork for more advanced topics like series and mathematical induction.

Term Calculation

Calculating terms in a sequence with a recursive formula involves a repetitive process. You'll need to rely on arithmetic operations and, importantly, the results from previous calculations. In this exercise, each subsequent term was determined by the recursive formula provided, often involving substitution of known values to find an unknown term.Let's break down the calculation steps:

• Start with known values for the initial terms, such as \(a_1 = -1\) and \(a_2 = 5\).
• Apply the recursive formula to determine each next term. For example, \(a_3\) is computed from \(a_1\) and \(a_2\).
• Repeat the process, using two preceding terms to find the next term.

By methodically applying the recursive rule and ensuring all calculations are double-checked for accuracy, errors can be minimized. Mastery of term calculation using recursive formulas enhances problem-solving skills and primes students for tackling more complex sequences and recursion-based problems.