Problem 24 - Precalculus: Mathematical for Calculus

Question

Find the \(n\) th term of a sequence whose first several terms are given. $$-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \dots$$

Step-by-Step Solution

Answer

The formula for the nth term is \(a_n = (-1)^{n} \cdot \left(\frac{1}{3}\right)^n\).

Step 1: Identify the Pattern

Look at the sequence given: \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}\). We need to identify a pattern that describes the change from one term to the next. Notice that each term is obtained by multiplying the previous term by \(-\frac{1}{3}\).

Step 2: Determine the First Term and Common Ratio

The first term of the sequence, denoted as \(a\), is \(-\frac{1}{3}\). The common ratio, denoted as \(r\), is \(-\frac{1}{3}\). This is calculated based on the rule that each term is \(-\frac{1}{3}\) times the previous one.

Step 3: Write the Formula for the nth Term

For a geometric sequence, the \(n\)th term is given by the formula: \[ a_n = a \cdot r^{n-1} \]Substituting the known values, we have: \[ a_n = \left(-\frac{1}{3}\right) \cdot \left(-\frac{1}{3}\right)^{n-1} \]

Step 4: Simplify the Expression for the nth Term

Using the formula from Step 3, simplify:\[ a_n = (-1)^{n} \cdot \left(\frac{1}{3}\right)^n \].This accounts correctly for the alternating signs and the pattern of the numerators and denominators.

Key Concepts

nth term

In a geometric sequence, the "nth term" refers to a term located in the sequence at position 'n'. Each term in the sequence is defined by its position, making it unique and predictable. Understanding the nth term helps you identify any term in the sequence without listing all prior terms.

To find the nth term, you start with the first term of the sequence, which is often denoted by 'a'. Given our sequence, the first term is \(-\frac{1}{3}\). You then multiply this term by a factor called the common ratio, raised to the power of \(n-1\), where 'n' is the term number. This is because the first term does not require multiplication by the common ratio.

Knowing how to find the nth term is crucial for sequences that follow a specific pattern, such as growth or decay sequences. In our sequence, the ability to find any term quickly makes calculations more efficient, especially when dealing with large 'n'. It also highlights the repeating pattern that characterizes geometric sequences.

common ratio

The "common ratio" is a core feature of any geometric sequence. It is the factor that you multiply by to get from one term to the next. In the exercise sequence \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}\), the common ratio is \(-\frac{1}{3}\).

You can find the common ratio by dividing any term by the term before it. For example, dividing \(\frac{1}{9}\) by \(-\frac{1}{3}\) will yield \(-\frac{1}{3}\). This pattern applies to any consecutive terms in the sequence. The ratio remains constant throughout, meaning every term adjusts by the same multiplier.

• The common ratio can be positive or negative.
• If the ratio is positive, the sequence maintains the same sign as it progresses.
• If negative, as with our sequence, the terms alternate in sign.

Understanding the common ratio provides insight into the behavior of the sequence. It affects the rapidity and direction of growth or decay and is essential for applying the geometric sequence formula effectively.

geometric sequence formula

The geometric sequence formula is fundamental for calculating any term in the sequence using the known parameters. The generic form of the geometric sequence formula is given by:

• \[ a_n = a \cdot r^{n-1} \]

Here, \(a\) is the first term, and \(r\) is the common ratio. The power \(n-1\) adjusts the formula so the first term remains as \(a\) without multiplying by the common ratio.

In our provided example, plugging in \(a = -\frac{1}{3}\) and \(r = -\frac{1}{3}\), we derive the formula:

• \[ a_n = \left(-\frac{1}{3}\right) \cdot \left(-\frac{1}{3}\right)^{n-1} \]

Upon simplifying further, this becomes:

• \[ a_n = (-1)^n \cdot \left(\frac{1}{3}\right)^n \]

The formula captures both the alternating sign and the consistent ratio as the sequence expands. With the geometric sequence formula, predicting terms becomes systematic and efficient, allowing you to skip directly to any desired nth term.