Problem 23 - College Algebra

Question

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$0.0004,-0.004,0.04,-0.4, \dots$$

Step-by-Step Solution

Answer

The general term of the sequence is \(a_n = 0.0004 * (-10^{n-1})\) and the seventh term of the sequence is -400.

Step 1: Find the common ratio

The common ratio \(r\) in a geometric sequence can be found by dividing any term by the preceding term. Here \(r = -0.004/0.0004 = -10.\)

Step 2: Write the formula for general term

The nth term of a geometric sequence is given by \(a_n = a_1 * (r^{n-1})\), where \(a_n\) is the nth term, \(a_1\) is the first term and \(r\) is the common ratio. Here, the first term \(a_1 = 0.0004\) and the common ratio \(r = -10\). So, the general term is \(a_n = 0.0004 * (-10^{n-1}).\)

Step 3: Find the seventh term

Substitute \(n = 7\) into the general term \(a_n = 0.0004 * (-10^{n-1})\) to find out the seventh term \(a_7.\) So, \(a_7 = 0.0004 * (-10^{6}) = -400.\)

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the **common ratio** is a crucial element that determines the pattern of the sequence. It's the factor by which we multiply a term to get the subsequent term. To find the common ratio, simply divide any term in the sequence by its preceding term.
For example, in the sequence given: 0.0004, -0.004, 0.04, -0.4,..., the common ratio \( r \) is achieved by taking \(-0.004 \div 0.0004 = -10\). Every single term increases or decreases by this factor. This consistency in multiplication helps define the nature and behavior of the entire sequence.
Key Points:

• A positive common ratio keeps the sequence terms in the same sign.
• A negative common ratio flips the sign of terms, resulting in alternating signs as seen in this problem.

Using the Nth Term Formula

The **nth term formula** is a vital tool for defining the position of any specific term within a geometric sequence. This formula is expressed as \( a_n = a_1 \times (r^{n-1}) \), where:

• \( a_n \) is the term you are calculating.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number.

With the formula, you can pinpoint any term in the sequence without listing all the previous terms. In our example sequence, the formula becomes \( a_n = 0.0004 \times (-10^{n-1}) \). This provides a quick method to calculate any desired term.

Delving into Sequence Analysis

**Sequence analysis** is all about understanding the characteristics and patterns of sequences, particularly geometric sequences. When analyzing a sequence, you determine properties like growth pattern, behavior of terms, and overall trend through its formula.
For the provided sequence, using the common ratio and nth term formula, it becomes evident that the sequence exhibits exponential growth. Each term is essentially the previous term multiplied by \(-10\). This reveals a pattern of rapid growth in magnitude and alternating sign.
**Important Observations:**

• Geometric sequences can model exponential growth or decay, identifiable from the common ratio.
• The absolute value of the common ratio over 1 indicates exponential growth, while less than 1 means decay.
• In this sequence's case, the alternating sign gives a curious swing between positive and negative values.

Steps in Term Calculation

**Term calculation** in a geometric sequence involves substituting the term position \( n \) into the nth term formula to find the term's value. In this instance, we already possess the nth term formula \( a_n = 0.0004 \times (-10^{n-1}) \).
To find the seventh term \( a_7 \):

• Start with substituting \( n = 7 \) into the formula.
• The expression transforms to \( a_7 = 0.0004 \times (-10^6) \).
• This simplifies to \( a_7 = 0.0004 \times -1,000,000 \).
• The calculated seventh term is \( -400 \).

By taking these steps, you can efficiently determine any term’s value, leveraging the sequence's inherent pattern.