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. $$0.0004,-0.004,0.04,-0.4, \ldots$$

Step-by-Step Solution

Answer

The seventh term \(a_{7}\) of the given geometric sequence is -400.

Step 1: Determine the common ratio

Observe the given sequence \(0.0004,-0.004,0.04,-0.4... \). Notice that each term is 10 times the preceding term, albeit with alternating signs. Therefore, the common ratio \(r\) in this geometric sequence is -10.

Step 2: Write out the general term formula

The formula for 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, \(r\) is the common ratio, and \(n\) is the term number.

Step 3: Substitute the known values into the formula

Substitute the given first term \(a_{1} = 0.0004\), and the determined ratio \(r = -10\), into the formula for the seventh term \(a_{7}\). Therefore \(a_{7} = 0.0004 * (-10)^{7-1}\).

Step 4: Simplify the expression

On simplifying the expression, since anything raised to an even power is positive, the term \(a_{7}\) becomes negative as it's multiplied by -10 six times, which leaves a positive product, and then multiplied by -10 once more, resulting in -400.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that helps in understanding how the sequence progresses. The common ratio is the factor by which each term in the sequence is multiplied to obtain the next term. This ratio is constant throughout the entire sequence.
To find the common ratio, you divide any term in the sequence by the previous term. For the sequence given, which is \(0.0004, -0.004, 0.04, -0.4, \ldots\), you can calculate:

• Divide \(-0.004\) by \(0.0004\)
• Or, divide \(0.04\) by \(-0.004\)

This yields a common ratio of \(-10\). It's important to note how the sign changes, also indicating that the sequence alternates between positive and negative values.

nth Term

The nth term of a geometric sequence represents the general form of any term in the sequence based on its position number \(n\). This term can be derived once you know the first term and the common ratio.
For the sequence given, the first term \(a_1\) is \(0.0004\) and the common ratio \(r\) is \(-10\). To find the nth term, you use the formula:

• \(a_{n} = a_1 \times r^{(n-1)}\)

In this classic step, you can replace \(a_1\) with \(0.0004\) and \(r\) with \(-10\) to express the nth term.

General Term Formula

The general term formula for a geometric sequence makes it easy to find any term in the sequence without listing all the preceding terms. This formula is very powerful especially for larger sequences.
The general term is expressed as:

• \(a_{n} = a_1 \times r^{(n-1)}\)

Here, \(a_1\) is the initial term and \(r\) is the common ratio.
The significance of this formula lies in its simplicity and efficiency, allowing for direct calculation of any term, such as \(a_7\), by substituting the values directly into the formula. For example, \(a_7\) in the example given would be calculated as \(0.0004 \times (-10)^6\).

Sequence Analysis

Analyzing a sequence involves understanding its pattern and behavior over a set number of terms. This involves checking the sequence's characteristics such as its sign, value growth or decay, and compact representation using formulas.
For the example \(0.0004, -0.004, 0.04, -0.4, \ldots\), the sequence alternates in sign, suggesting a pattern of positive to negative values. Identifying such patterns helps in predicting future terms.
Sequence analysis often involves:

• Identifying clear patterns or anomalies
• Calculating specific terms using generalized formulas
• Verifying sequence behavior mathematically

Through such analysis, attributes like convergence or divergence can also be determined, enriching the understanding of the sequence beyond immediate term calculations.