Problem 17 - College Algebra

Question

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} \frac{n^{2}}{2 n+1} \text { if } n \leq 5 \\ n^{2}-5 \text { if } n>5 \end{array}\right. $$

Step-by-Step Solution

Answer

The first eight terms are \( \frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \frac{25}{11}, 31, 44, 59 \).

Step 1: Understand the Problem

We are given a piecewise sequence where the sequence definition changes based on the value of n. For \( n \leq 5 \), use \( \frac{n^2}{2n+1} \). For \( n > 5 \), use \( n^2 - 5 \). We need to find the first eight terms \( a_1, a_2, \ldots, a_8 \).

Step 2: Calculate Terms for \( n \leq 5 \)

Calculate the terms for when \( n \leq 5 \): - For \( n = 1 \), \( a_1 = \frac{1^2}{2(1)+1} = \frac{1}{3} \) - For \( n = 2 \), \( a_2 = \frac{2^2}{2(2)+1} = \frac{4}{5} \) - For \( n = 3 \), \( a_3 = \frac{3^2}{2(3)+1} = \frac{9}{7} \) - For \( n = 4 \), \( a_4 = \frac{4^2}{2(4)+1} = \frac{16}{9} \) - For \( n = 5 \), \( a_5 = \frac{5^2}{2(5)+1} = \frac{25}{11} \).These calculations provide the terms from \( a_1 \) to \( a_5 \).

Step 3: Calculate Terms for \( n > 5 \)

Now calculate terms for when \( n > 5 \): - For \( n = 6 \), \( a_6 = 6^2-5 = 31 \) - For \( n = 7 \), \( a_7 = 7^2-5 = 44 \) - For \( n = 8 \), \( a_8 = 8^2-5 = 59 \).These calculations provide terms from \( a_6 \) to \( a_8 \).

Step 4: Compile the Sequence

Now that we have both sections of the sequence calculated, let's compile them together:- \( a_1 = \frac{1}{3} \)- \( a_2 = \frac{4}{5} \)- \( a_3 = \frac{9}{7} \)- \( a_4 = \frac{16}{9} \)- \( a_5 = \frac{25}{11} \)- \( a_6 = 31 \)- \( a_7 = 44 \)- \( a_8 = 59 \).Thus, the first eight terms of the sequence are \( \frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \frac{25}{11}, 31, 44, 59 \).

Key Concepts

Terms Calculation

Calculating the terms of a piecewise sequence involves evaluating the sequence definition for specific values of \( n \). Each term of the sequence corresponds to a certain natural number starting from 1, i.e., \( n = 1, 2, 3, \ldots \). For such a sequence, different formulas might be used depending on the range of \( n \):

• For \( n \leq 5 \), we use the formula \( a_n = \frac{n^2}{2n+1} \)
• For \( n > 5 \), we use the formula \( a_n = n^2 - 5 \)

These formulas are evaluated by substituting the specified values for \( n \) within their respective domains. The terms for \( n = 1 \) to \( n = 5 \) are calculated using the first formula, which involves polynomial division. The terms for \( n = 6 \) to \( n = 8 \) use a straightforward quadratic polynomial. This method of calculation ensures that each term appropriately fits the piecewise nature of the sequence.

Sequence Definition

In mathematics, sequence definition is crucial for understanding the kind of sequence you are dealing with and how to determine its terms. A sequence is generally an ordered list of numbers or objects, and in this context, we explore piecewise sequences.Piecewise sequences are defined by different expressions based on the value of \( n \). For example, our sequence is defined differently when \( n \leq 5 \) than when \( n > 5 \). This is common in piecewise definitions where the sequence changes rules or formula depending on whether \( n \) is within a certain range:

• If \( n \leq 5 \), the sequence is defined as \( a_n = \frac{n^2}{2n+1} \)
• If \( n > 5 \), it changes to \( a_n = n^2 - 5 \)

Understanding this piecewise definition allows us to accurately compute each needed term and analyze how they relate to one another in broader mathematical reasoning.

Mathematical Sequences

Mathematical sequences are fundamental concepts in mathematics that represent a structured set of elements, usually numbers, arranged in a particular order. They provide a framework to understand sequences’ behavior, growth patterns, and potential rules governing subsequent numbers. One key characteristic of a sequence is that each term is generated using a defined rule, which may be mathematical expressions or logical rules. These rules dictate how each term relates to the others and can vary widely from one sequence to another. Our piecewise sequence is an example of how mathematical sequences can dynamically change rules across positions. Piecewise sequences are powerful tools in modeling real-world scenarios where behavior changes at certain thresholds. For example, electricity tariff calculations often change when the consumption reaches certain levels, much like a piecewise sequence does with \( n \). By understanding how they work, students can better grasp not only mathematical sequences but also their diverse applications in various fields.