Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, write the first eight terms of the piecewise sequence. \(a_{n}=\left\\{\begin{array}{ll}4\left(n^{2}-2\right) & \text { if } n \leq 3 \text { or } n>6 \\ \frac{n^{2}-2}{4} & \text { if } 3

Step-by-Step Solution

Answer

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

Step 1: Identify the formula for each range

The sequence is given in two parts:- Use the formula \( 4(n^2 - 2) \) for \( n \leq 3 \) or \( n > 6 \).- Use the formula \( \frac{n^2 - 2}{4} \) for \( 3 < n \leq 6 \).

Step 2: Calculate terms for \( n = 1, 2, 3 \)

Since \( n \leq 3 \), we use \( 4(n^2 - 2) \).- For \( n = 1 \), calculate \( 4(1^2 - 2) = 4(-1) = -4 \).- For \( n = 2 \), calculate \( 4(2^2 - 2) = 4(2) = 8 \).- For \( n = 3 \), calculate \( 4(3^2 - 2) = 4(7) = 28 \).

Step 3: Calculate terms for \( n = 4, 5, 6 \)

Since \( 3 < n \leq 6 \), we use \( \frac{n^2 - 2}{4} \).- For \( n = 4 \), calculate \( \frac{4^2 - 2}{4} = \frac{14}{4} = 3.5 \).- For \( n = 5 \), calculate \( \frac{5^2 - 2}{4} = \frac{23}{4} = 5.75 \).- For \( n = 6 \), calculate \( \frac{6^2 - 2}{4} = \frac{34}{4} = 8.5 \).

Step 4: Calculate terms for \( n = 7, 8 \)

Since \( n > 6 \), we use \( 4(n^2 - 2) \).- For \( n = 7 \), calculate \( 4(7^2 - 2) = 4(47) = 188 \).- For \( n = 8 \), calculate \( 4(8^2 - 2) = 4(62) = 248 \).

Step 5: Write the first eight terms

Based on the calculations: - Terms are: -4, 8, 28, 3.5, 5.75, 8.5, 188, 248.

Key Concepts

Sequence Calculation

Calculating the sequence in a systematic manner is crucial for evaluating each term accurately. In the exercise provided, a piecewise sequence is used. This means the formula changes based on the value of \( n \). When solving these types of problems, it's important to:

• Identify the formula used for different ranges of \( n \).
• Methodically compute each term by substituting the value of \( n \) into the correct formula.

To find the terms, substitute \( n = 1 \) through \( n = 8 \) into the specified equations. This approach ensures clarity and accuracy.

Mathematical Terms

Mathematical terms are the language through which complex ideas in math can be expressed and understood. In the context of sequences, some terms you should become familiar with include:

• Term: An individual element or number in a sequence.
• Range: Conditions set on the variable \( n \) that determine which formula to use.
• Piecewise: Defined by different expressions for different intervals of the input value.

These terms form the foundation of understanding how sequences and functions are constructed and evaluated.

Piecewise Function

A piecewise function is a function composed of multiple sub-functions, each applying to a certain interval of the main function's domain. For sequence calculations, this can be seen when different formulas are used based on the value of \( n \). Knowing how to:

• Identify the correct sub-function based on the input value.
• Apply the correct formula for each interval.

This is necessary for correctly calculating a piecewise sequence. In mathematical problems, understanding how to handle these conditions allows students to solve more complex and diverse problems confidently.