Problem 20 - College Algebra

Question

Write the first eight terms of the piecewise sequence. $$a_{n}=\left\\{\begin{array}{l}{4\left(n^{2}-2\right) \text { if } n \leq 3 \text { or } n > 6} \\ {\frac{n^{2}-2}{4} \text { if } 3 < n \leq 6}\end{array}\right.$$

Step-by-Step Solution

Answer

The sequence terms are: -4, 8, 28, 3.5, 5.75, 8.5, 188, 248.

Step 1: Identify the conditions for applying the formulas

The sequence is defined using two different formulas based on the value of \( n \). The first formula \( a_n = 4(n^2 - 2) \) is for when \( n \leq 3 \) or \( n > 6 \). The second formula \( a_n = \frac{n^2 - 2}{4} \) is for \( 3 < n \leq 6 \). We'll use these formulas to generate the first eight terms.

Step 2: Calculate terms when \( n \leq 3 \)

Using the formula \( a_n = 4(n^2 - 2) \), calculate the first three terms:\- When \( n = 1 \): \( a_1 = 4(1^2 - 2) = 4(-1) = -4 \)\- When \( n = 2 \): \( a_2 = 4(2^2 - 2) = 4(2) = 8 \)\- When \( n = 3 \): \( a_3 = 4(3^2 - 2) = 4(7) = 28 \)

Step 3: Calculate terms for \( 3 < n \leq 6 \)

For this range, use the formula \( a_n = \frac{n^2 - 2}{4} \):\- When \( n = 4 \): \( a_4 = \frac{4^2 - 2}{4} = \frac{16 - 2}{4} = \frac{14}{4} = 3.5 \)\- When \( n = 5 \): \( a_5 = \frac{5^2 - 2}{4} = \frac{25 - 2}{4} = \frac{23}{4} = 5.75 \)\- When \( n = 6 \): \( a_6 = \frac{6^2 - 2}{4} = \frac{36 - 2}{4} = \frac{34}{4} = 8.5 \)

Step 4: Calculate terms when \( n > 6 \)

Since \( n \) is greater than 6, revert to the formula \( a_n = 4(n^2 - 2) \):\- When \( n = 7 \): \( a_7 = 4(7^2 - 2) = 4(47) = 188 \)\- When \( n = 8 \): \( a_8 = 4(8^2 - 2) = 4(62) = 248 \)

Step 5: List the first eight terms of the sequence

Combine the results from previous steps to list the sequence:\- \( a_1 = -4 \)\- \( a_2 = 8 \)\- \( a_3 = 28 \)\- \( a_4 = 3.5 \)\- \( a_5 = 5.75 \)\- \( a_6 = 8.5 \)\- \( a_7 = 188 \)\- \( a_8 = 248 \)

Key Concepts

Understanding Algebraic Sequences

Algebraic sequences are mathematical patterns where terms are defined by algebraic expressions. They follow rules allowing calculation from term to term. In this case, our sequence is piecewise, implying different algebraic expressions apply to various sections of the sequence. Students often encounter conditions within an algebraic sequence such as:

• Values of \( n \) less than or equal to a specific number.
• Values of \( n \) greater than a specified number.
• Values of \( n \) between two numbers.

Recognizing these conditions helps decide which expression to use. It is crucial because each expression impacts terms differently depending on its condition. Piecewise sequences introduce a layered complexity, requiring attention to detail as you swap between expressions based on the value of \( n \). A clear understanding of these concepts not only improves one's handling of algebraic sequences but also fosters confidence in tackling more complex mathematics problems.

Mathematics Problem Solving Techniques for Sequences

Problem solving in mathematics is about methodically breaking down problems into manageable parts. In a sequence scenario, like the piecewise sequence, it's essential first to understand the given conditions then identify which formula applies to each segment.

When dealing with sequences, particularly piecewise sequences, you can harness several effective techniques:

• Clearly distinguish conditions to know which formula to apply for different values of \( n \).
• Solve for each condition one step at a time to avoid confusion.
• Ensure calculations are accurate by double-checking each arithmetic operation within the expressions.

Navigating through differentiating conditions is a crucial skill here. This methodical approach facilitates orderly and correct substitutions of numbers for \( n \), ensuring each term is computed accurately. Being systematic also minimizes errors, a common pitfall in sequences involving various conditions.

Calculating Sequence Terms

Calculating terms of a sequence, especially in piecewise scenarios, requires following rigorous steps to derive each term accurately. Each formula in a piecewise sequence encodes specific instructions that depend on the value \( n \) takes in the sequence.

In this piecewise sequence, we first identify which formula to use:

• For terms where \( n \) is less than or equal to 3 or greater than 6, use \( a_n = 4(n^2 - 2) \).
• For terms where \( 3 < n \leq 6 \), switch to \( a_n = \frac{n^2 - 2}{4} \).

As we compute each term step by step:

• Substitute the specific \( n \) into the chosen expression.
• Compute the square, then apply other operations - multiplication, division, addition, or subtraction, as specified.
• Ensure accuracy to keep your sequence consistent and correct.

This attention to detail, ensuring computations align with the conditions, is critical to avoid errors. Practicing this improves your ability to work with sequences and enhances your overall problem-solving skills in mathematics.