Problem 20 - College Algebra

Question

For the following exercises, 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

Step-by-Step Solution

Answer

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

Step 1: Determine for Each n Which Formula to Use

For the sequence given, identify which formula to apply based on the value of \( n \):- For \( n \leq 3 \) or \( n > 6 \), use \( a_n = 4(n^2 - 2) \).- For \( 3 < n \leq 6 \), use \( a_n = \frac{n^2 - 2}{4} \).

Step 2: Calculate First Term (n=1)

For \( n = 1 \), since \( n \leq 3 \):\[a_1 = 4(1^2 - 2) = 4(-1) = -4.\]

Step 3: Calculate Second Term (n=2)

For \( n = 2 \), since \( n \leq 3 \):\[a_2 = 4(2^2 - 2) = 4(4 - 2) = 4 \times 2 = 8.\]

Step 4: Calculate Third Term (n=3)

For \( n = 3 \), since \( n \leq 3 \):\[a_3 = 4(3^2 - 2) = 4(9 - 2) = 4 \times 7 = 28.\]

Step 5: Calculate Fourth Term (n=4)

For \( n = 4 \), since \( 3 < n \leq 6 \):\[a_4 = \frac{4^2 - 2}{4} = \frac{16 - 2}{4} = \frac{14}{4} = 3.5.\]

Step 6: Calculate Fifth Term (n=5)

For \( n = 5 \), since \( 3 < n \leq 6 \):\[a_5 = \frac{5^2 - 2}{4} = \frac{25 - 2}{4} = \frac{23}{4} = 5.75.\]

Step 7: Calculate Sixth Term (n=6)

For \( n = 6 \), since \( 3 < n \leq 6 \):\[a_6 = \frac{6^2 - 2}{4} = \frac{36 - 2}{4} = \frac{34}{4} = 8.5.\]

Step 8: Calculate Seventh Term (n=7)

For \( n = 7 \), since \( n > 6 \):\[a_7 = 4(7^2 - 2) = 4(49 - 2) = 4 \times 47 = 188.\]

Step 9: Calculate Eighth Term (n=8)

For \( n = 8 \), since \( n > 6 \):\[a_8 = 4(8^2 - 2) = 4(64 - 2) = 4 \times 62 = 248.\]

Key Concepts

sequence terms calculation

When working with piecewise sequences, the first step is to understand how to identify which formula applies to each term. In these sequences, different formulas are applied to different segments of the sequence depending on the value of the term's position, denoted as \( n \). Here’s how you calculate the terms:

• First, determine if the condition \( n \leq 3 \) or \( n > 6 \) is true; in this case, use the formula \( a_n = 4(n^2 - 2) \).
• If not, check whether the condition \( 3 < n \leq 6 \) holds; then use \( a_n = \frac{n^2 - 2}{4} \).

Knowing which segment \( n \) falls into guides you to apply the correct formula and calculate each term accordingly, as shown in the step-by-step solution above.

formula application

Once you know which segment \( n \) falls into for a piecewise sequence, applying the formula becomes straightforward. Let’s break down the application process:
For the term calculation:

• Plug the value of \( n \) into the correct formula determined by its range.
• For the equation \( a_n = 4(n^2 - 2) \): calculate \( n^2 \), subtract 2, multiply the result by 4.
• For \( a_n = \frac{n^2 - 2}{4} \): calculate \( n^2 \), subtract 2, and then divide the result by 4.

Focus on applying these mathematical operations in sequence to arrive at the correct term value for each \( n \). Errors often happen if terms of the sequence are calculated using the formula mistaken for another segment, hence carefully observe the conditions prior to calculation.

mathematical sequences

Mathematical sequences, particularly piecewise sequences, are structured lists of numbers that follow a specific order and formula rules. These sequences rely heavily on defined conditions and formulas to generate each term. Here's what you should know:

• A sequence term is generated based on its position \( n \), using predetermined formulas.
• The term for each \( n \) may change if \( n \) crosses into a different range (or segment) as defined by the piecewise conditions.
• Understand that a sequence can have infinite terms, but you typically calculate only those needed or specified by a problem.

Such sequences are crucial in mathematics for modeling and solving real-world problems where different conditions lead to different outcomes. Piecewise sequences make it possible to address these varied patterns effectively.