Problem 10 - Algebra for College Students

Question

List all terms of each finite sequence. \(b_{n}=\frac{(-1)^{n+1}}{n}\) for \(1 \leq n \leq 6\)

Step-by-Step Solution

Answer

The terms are: 1, -1/2, 1/3, -1/4, 1/5, -1/6.

Step 1: Understand the Sequence Formula

Identify the given formula for the sequence: \(b_{n}=\frac{(-1)^{n+1}}{n}\). This formula shows how each term in the sequence is calculated based on the term number \(n\).

Step 2: Calculate the First Term

Substitute \(n = 1\) into the formula: \(b_{1}=\frac{(-1)^{1+1}}{1}=\frac{1}{1}=1\). The first term is 1.

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula: \(b_{2}=\frac{(-1)^{2+1}}{2}=\frac{-1}{2}\). The second term is -1/2.

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula: \(b_{3}=\frac{(-1)^{3+1}}{3}=\frac{1}{3}\). The third term is 1/3.

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \(b_{4}=\frac{(-1)^{4+1}}{4}=\frac{-1}{4}\). The fourth term is -1/4.

Step 6: Calculate the Fifth Term

Substitute \(n = 5\) into the formula: \(b_{5}=\frac{(-1)^{5+1}}{5}=\frac{1}{5}\). The fifth term is 1/5.

Step 7: Calculate the Sixth Term

Substitute \(n = 6\) into the formula: \(b_{6}=\frac{(-1)^{6+1}}{6}=\frac{-1}{6}\). The sixth term is -1/6.

Key Concepts

Understanding the Sequence Formula

A sequence formula defines how each term in a sequence is calculated. In our exercise, the formula is given as: \[ b_{n} = \frac{(-1)^{n+1}}{n} \].
This means that every term of the sequence is determined by the term number, denoted as \(n\).
The formula involves alternating signs due to the power of \((-1)\). When \(n\) is odd, \((-1)^{n+1}\) becomes positive, and when \(n\) is even, \((-1)^{n+1}\) becomes negative.
This guides us in different terms of the sequence.

Term Calculation

To find the terms of the sequence, we simply substitute different values of \(n\) into the formula.
For instance:

• For \(n = 1\), we substitute into the formula to get the first term: \[ b_{1} = \frac{(-1)^{1+1}}{1} = 1 \]
  Therefore, the first term is 1.
• For \(n = 2\), we find the second term: \[ b_{2} = \frac{(-1)^{2+1}}{2} = \frac{-1}{2} \]
  So, the second term is -1/2.
• For \(n = 3\), the third term would be: \[ b_{3} = \frac{(-1)^{3+1}}{3} = \frac{1}{3} \]
  Hence, the third term is 1/3.
• Continuing in this pattern, we calculate as follows:
  For \(n = 4\): \[ b_{4} = \frac{-1}{4} \],
  For \(n = 5\): \[ b_{5} = \frac{1}{5} \],
  For \(n = 6\): \[ b_{6} = \frac{-1}{6} \]

This pattern shows how each term can be calculated step-by-step by substituting the value of \(n\).

Finite Sequence

A sequence can be either finite or infinite. When a sequence has a specific number of terms, it is called a finite sequence.
In our exercise, the given sequence extends from \(n = 1\) to \(n = 6\). Therefore, it is finite, consisting of six terms.
This starts us off on understanding finite sequences, where there is a clear beginning and end.