Problem 6 - College Algebra

Question

Write the first four terms of each sequence whose general term is given. $$a_{n}-\left(-\frac{1}{3}\right)^{n}$$

Step-by-Step Solution

Answer

The first four terms of the sequence are \( \frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}\).

Step 1: Understand the General Term

The general term \(-\frac{1}{3}^n\) defines any term of the sequence. Here, 'n' is the term number. To find the term of the sequence, 'n' in the general term is replaced by the term number.

Step 2: Substitution for the First Term

To find the first term of the sequence, replace 'n' in the general term by 1. Hence, \(a_1=-\frac{1}{3}^1 = \frac{1}{3}\).

Step 3: Substitution for the Second Term

To find the second term of the sequence, replace 'n' in the general term by 2. Hence, \(a_2 = -\frac{1}{3}^2 = \frac{1}{9}\).

Step 4: Substitution for the Third Term

To find the third term of the sequence, replace 'n' in the general term by 3. Hence, \(a_3 = -\frac{1}{3}^3 = -\frac{1}{27}\).

Step 5: Substitution for the Fourth Term

To find the fourth term of the sequence, replace 'n' in the general term with 4. Hence, \(a_4 = -\frac{1}{3}^4 = \frac{1}{81}\).

Key Concepts

Understanding the General Term

A sequence is often defined by a **general term** that represents each term in a sequence based on its position. In our example, the general term is given by the formula \( a_n = -\left(\frac{1}{3}\right)^n \). Here, \( n \) stands for the term's position in the sequence. Knowing the general term is key because it helps you find any term in your sequence by simply plugging in the number of the term into this formula.

• For instance, \( n=1 \) would give you the first term.
• When \( n=2 \), you find the second term, and so on.

Understanding this formula means understanding how every term in your sequence will behave as \( n \) changes. It serves as a shortcut to calculating terms in the sequence without having to compute all preceding terms.

Substitution to Find Sequence Terms

**Substitution** is the next step after understanding your general term. This process involves replacing \( n \) in your general term with different integers to find specific sequence terms. For our sequence, you substitute \( n \) with values like 1, 2, 3, etc.
Here's how it typically works:

• Substitute \( n=1 \) in the general term \( a_n = -\left(\frac{1}{3}\right)^n \) to find the first term: \( a_1 = -\left(\frac{1}{3}\right)^1 = \frac{1}{3} \).
• Continue by substituting \( n=2 \) for the second term: \( a_2 = -\left(\frac{1}{3}\right)^2 = \frac{1}{9} \).
• Keep substituting for more terms (\( n=3, 4, ... \)). For instance, \( a_3 = -\left(\frac{1}{3}\right)^3 = -\frac{1}{27} \).

By following substitution, you can systematically and easily find any term in the sequence.

Finding and Understanding Sequence Terms

Once you substitute the values of \( n \) to the general term, you uncover the **sequence terms**. These are the individual terms of your sequence, and each term has unique characteristics. For example, the terms we calculated from the general term \( a_n = -\left(\frac{1}{3}\right)^n \) are as follows:

• The first term is \( a_1 = \frac{1}{3} \).
• The second term is \( a_2 = \frac{1}{9} \).
• The third term is \( a_3 = -\frac{1}{27} \).
• The fourth term is \( a_4 = \frac{1}{81} \).

These terms showcase how each position in the sequence is defined by the formula, revealing a pattern or behavior. Identifying and understanding these sequence terms helps you predict subsequent terms and understand the overarching pattern or rule the sequence follows over time.