Problem 21 - Algebra for College Students

Question

Write the first four terms of the infinite sequence whose nth term is given. \(b_{n}=\frac{1}{2 n-5}\)

Step-by-Step Solution

Answer

The first four terms are \( -\frac{1}{3}, -1, 1, \frac{1}{3} \).

Step 1 - Identify the general term

The general term of the sequence is given by the formula: \[ b_{n} = \frac{1}{2n - 5} \]

Step 2 - Calculate the first term

To find the first term of the sequence, substitute \(n = 1\) into the general term: \[ b_{1} = \frac{1}{2(1) - 5} = \frac{1}{2 - 5} = \frac{1}{-3} = -\frac{1}{3} \]

Step 3 - Calculate the second term

To find the second term of the sequence, substitute \(n = 2\) into the general term: \[ b_{2} = \frac{1}{2(2) - 5} = \frac{1}{4 - 5} = \frac{1}{-1} = -1 \]

Step 4 - Calculate the third term

To find the third term of the sequence, substitute \(n = 3\) into the general term: \[ b_{3} = \frac{1}{2(3) - 5} = \frac{1}{6 - 5} = \frac{1}{1} = 1 \]

Step 5 - Calculate the fourth term

To find the fourth term of the sequence, substitute \(n = 4\) into the general term: \[ b_{4} = \frac{1}{2(4) - 5} = \frac{1}{8 - 5} = \frac{1}{3} \]

Key Concepts

General Term

To fully understand infinite sequences, we start with the general term. The general term is a formula that provides a rule for finding any term in the sequence. In this exercise, the general term is given as: \[ b_{n} = \frac{1}{2n - 5} \] This formula means that for any positive integer value of \( n \), you can substitute \( n \) in this equation to get the \( n \)-th term of the sequence. Understanding the general term is key to working with sequences, as it simplifies the process of finding individual terms.

Substitution

Substitution is the process of replacing a variable with a number or another expression. When working with sequences, we substitute the position of the term (like \( n \)) into the general term. This helps us calculate specific terms. To find the first term \( b_{1} \), substitute \( n = 1 \): \[ b_{1} = \frac{1}{2(1) - 5} = \frac{1}{2 - 5} = \frac{1}{-3} = -\frac{1}{3} \] Follow similar steps to find the next terms. Substituting different values of \( n \) lets you calculate as many terms of the sequence as you need.

Sequence Terms

Sequence terms are the individual numbers in a sequence. Given the general term \( b_{n} = \frac{1}{2n - 5} \), we can use substitution to find the first four terms:

• First term (\(n = 1\)): \( \frac{1}{-3} \)
• Second term (\( n = 2 \)): \( -1 \)
• Third term (\( n = 3\)): \( 1 \)
• Fourth term (\( n = 4 \)): \( \frac{1}{3} \)

Each of these terms result from substituting the position number (\( n \)) into the general term formula and performing arithmetic calculations.

Arithmetic Calculations

Arithmetic calculations involve basic mathematical operations like addition, subtraction, multiplication, and division. To find the sequence terms, perform arithmetic calculations:

• Identify the value for \( n \) to determine which term you're calculating.
• Substitute this value into the general term formula.
• Simplify the equation using arithmetic operations.

For example, let's calculate the second term with \( n = 2 \): \[ b_{2} = \frac{1}{2(2) - 5} = \frac{1}{4 - 5} = \frac{1}{-1} = -1 \] Repeating these steps helps you understand and find any term in the sequence. Arithmetic calculations are essential for simplifying and interpreting results.