Problem 17 - College Algebra

Question

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 Sequence Definition

The sequence is defined piecewise, meaning it has different expressions based on the value of \(n\). Specifically, for \(n \leq 5\), the term is given by \(\frac{n^2}{2n+1}\). For \(n > 5\), the term is \(n^2 - 5\). We need to calculate the first eight terms where \(n\) ranges from 1 to 8.

Step 2: Calculate Terms for n ≤ 5

For \(n = 1\) to \(n = 5\), use the formula \(\frac{n^2}{2n+1}\):- \(a_1 = \frac{1^2}{2 \times 1 + 1} = \frac{1}{3}\)- \(a_2 = \frac{2^2}{2 \times 2 + 1} = \frac{4}{5}\)- \(a_3 = \frac{3^2}{2 \times 3 + 1} = \frac{9}{7}\)- \(a_4 = \frac{4^2}{2 \times 4 + 1} = \frac{16}{9}\)- \(a_5 = \frac{5^2}{2 \times 5 + 1} = \frac{25}{11}\)

Step 3: Evaluate Terms for n > 5

For \(n = 6\) to \(n = 8\), use the formula \(n^2 - 5\):- \(a_6 = 6^2 - 5 = 36 - 5 = 31\)- \(a_7 = 7^2 - 5 = 49 - 5 = 44\)- \(a_8 = 8^2 - 5 = 64 - 5 = 59\)

Step 4: List the First Eight Terms

Combine all the calculated terms to form the first eight terms of the sequence:- \(a_1 = \frac{1}{3}\)- \(a_2 = \frac{4}{5}\)- \(a_3 = \frac{9}{7}\)- \(a_4 = \frac{16}{9}\)- \(a_5 = \frac{25}{11}\)- \(a_6 = 31\)- \(a_7 = 44\)- \(a_8 = 59\).

Key Concepts

first eight terms

When working with sequences, especially piecewise sequences, the objective is often to determine the initial terms. In this case, we want to find the first eight terms of the provided piecewise sequence.

• The sequence is defined with different rules depending on whether the index, represented by \(n\), is less than or equal to 5 or greater than 5.
• These terms are crucial because they show the pattern and behavior at the start of the sequence.

To find these terms, begin with \(n = 1\) up to \(n = 8\). For this particular sequence:

• For \(n = 1\), the term is \(\frac{1}{3}\).
• Continue calculating until you reach \(n = 5\), still using the same formula.
• For terms where \(n > 5\) (i.e., \(n = 6, 7, 8\)), use the second formula to find \(a_6 = 31\), \(a_7 = 44\), and \(a_8 = 59\).

Listing these, the first eight terms of the sequence are: \(\frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \frac{25}{11}, 31, 44,\) and \(59\). Understanding these initial terms is valuable as they lay the groundwork for identifying trends or behaviors in the sequence.

sequence definition

A sequence is a list of numbers in a specific order where each number is called a term. In mathematics, sequences are often defined by a rule that relates one term to its position in the sequence or to other terms. Here, we're dealing with a piecewise sequence.
A piecewise sequence means that the sequence is defined by different expressions depending on the value of the index \(n\).

• This sequence is crafted with two formulas: \(\frac{n^2}{2n+1}\) for \(n \leq 5\) and \(n^2 - 5\) for \(n > 5\).
• The distinction in these expressions exemplifies how piecewise sequences can adapt their definition based on different intervals of \(n\).

The fundamental purpose of having such a definition is to accurately describe behaviors under different conditions. By incorporating variable dependency, piecewise sequences provide the flexibility needed in real-world scenarios where a single formula might not suffice.

calculating sequence terms

To calculate terms in a piecewise sequence, you start by understanding the criteria that determine which formula to use for each term number \(n\). Here is a straightforward approach:

• Identify the range of \(n\) you want to consider. For example, focus on \(n = 1\) to \(n = 8\) for the initial terms.
• Apply the first formula \(\frac{n^2}{2n+1}\) whenever \(n \leq 5\).
• For example, calculate \(a_1\) as follows: substitute 1 into the formula to get \(\frac{1^2}{2 \times 1 + 1} = \frac{1}{3}\).
• Similarly, calculate \(a_2, a_3, a_4,\) and \(a_5\).
• Use the second formula \(n^2 - 5\) for \(n > 5\).
• For \(a_6\), \(a_7\), and \(a_8\), substitute the values 6, 7, and 8 respectively into the formula.

By carefully applying these calculations, you'll find the specific terms for each \(n\). This method ensures that each term reflects the exact rule established by the piecewise sequence definition.