Problem 18 - College Algebra

Question

Write the first eight terms of the piecewise sequence. $$a_{n}=\left\\{\begin{array}{l}{(2 n+1)^{2} \text { if } n \text { is divisible by } 4} \\ {\frac{2}{n} \text { if } n \text { is not divisible by } 4}\end{array}\right.$$

Step-by-Step Solution

Answer

The first eight terms are 2, 1, \(\frac{2}{3}\), 81, \(\frac{2}{5}\), \(\frac{1}{3}\), \(\frac{2}{7}\), and 289.

Step 1: Understand the Piecewise Sequence Definition

The sequence is defined in two parts based on the value of \(n\). If \(n\) is divisible by 4, then the term is calculated as \((2n+1)^2\). If \(n\) is not divisible by 4, then the term is calculated as \(\frac{2}{n}\).

Step 2: Calculate Terms for n=1 to n=4

1. For \(n=1\), 1 is not divisible by 4. Thus, \(a_1 = \frac{2}{1} = 2\).2. For \(n=2\), 2 is not divisible by 4. Thus, \(a_2 = \frac{2}{2} = 1\).3. For \(n=3\), 3 is not divisible by 4. Thus, \(a_3 = \frac{2}{3}\).4. For \(n=4\), 4 is divisible by 4. Thus, \(a_4 = (2 \times 4 + 1)^2 = 9^2 = 81\).

Step 3: Calculate Terms for n=5 to n=8

5. For \(n=5\), 5 is not divisible by 4. Thus, \(a_5 = \frac{2}{5}\).6. For \(n=6\), 6 is not divisible by 4. Thus, \(a_6 = \frac{2}{6} = \frac{1}{3}\).7. For \(n=7\), 7 is not divisible by 4. Thus, \(a_7 = \frac{2}{7}\).8. For \(n=8\), 8 is divisible by 4. Thus, \(a_8 = (2 \times 8 + 1)^2 = 17^2 = 289\).

Step 4: List the First Eight Terms

The first eight terms of the sequence are: \(a_1 = 2\), \(a_2 = 1\), \(a_3 = \frac{2}{3}\), \(a_4 = 81\), \(a_5 = \frac{2}{5}\), \(a_6 = \frac{1}{3}\), \(a_7 = \frac{2}{7}\), \(a_8 = 289\).

Key Concepts

sequence definition

A sequence is an ordered list of numbers following a specific rule. Each number in the list is called a term. In mathematics, a piecewise sequence is defined by different expressions, depending on certain conditions applied to the index, often noted as "n". For example, consider a sequence where its terms are defined differently if "n" is divisible by 4 and if "n" is not divisible by 4. This approach helps in defining sequences with varying behaviors across different ranges of index values. Understanding the definition of a sequence is crucial because it tells us how to determine each term based on its position.

divisibility

Divisibility is a concept in mathematics where one number can be divided by another without leaving a remainder. In sequence calculations, identifying whether the index "n" is divisible by a certain number (like 4 in our example) is essential. It dictates which piece of the piecewise function should be used to calculate the sequence terms. For instance, if "n" is divisible by 4, we use the expression \( (2n+1)^2 \) for the term. Divisibility here acts as the condition which decides the formula to apply, indicating its importance in determining the path of term calculation.

term calculation

Calculating terms in a sequence involves substituting the index value into the correct formula based on the defined rules. Let's go through the term calculation process for a piecewise sequence:

• If "n" is divisible by 4, use the formula \( (2n+1)^2 \).
• If "n" is not divisible by 4, use the formula \( \frac{2}{n} \).

For each "n" from 1 to 8, apply the appropriate formula determined by the divisibility rule. For instance, when calculating \( a_4 \), since 4 is divisible by 4, we compute \( (2 imes 4 + 1)^2 = 81 \). In another case, for \( a_2 \), since 2 is not divisible by 4, the term results in \( \frac{2}{2} = 1 \). Knowing the correct formula to apply directly impacts the calculated term.

mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a particular value or relationship. In a sequence, expressions help define how each term is generated. For our piecewise sequence, two expressions are used:

• \((2n+1)^2\) for terms where "n" is divisible by 4.
• \(\frac{2}{n}\) for terms where "n" is not divisible by 4.

These expressions provide a formulaic way to compute the sequence's terms based on the condition specified. Understanding how to read and apply these expressions in specific contexts is crucial. They serve as the building blocks for any sequence and dictate how it will be constructed numerically.