Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$1,-\frac{x}{3}, \frac{x^{2}}{9},-\frac{x^{3}}{27}, \ldots$$

Step-by-Step Solution

Answer

The nth term is \(a_n = \left(-\frac{x}{3}\right)^{n-1}\), the fifth term is \(\frac{x^4}{81}\), and the eighth term is \(-\frac{x^7}{2187}\).

Step 1: Identify the First Term and Common Ratio

The first term of the sequence is given as 1. To find the common ratio, divide the second term by the first term: \(-\frac{x}{3} \div 1 = -\frac{x}{3}\). Thus, the common ratio \(r = -\frac{x}{3}\).

Step 2: Write the General Formula for the nth Term

The general formula for the nth term of a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. For this sequence, \(a_n = 1 \cdot \left(-\frac{x}{3}\right)^{n-1}\).

Step 3: Find the Fifth Term

Replace \(n\) with 5 in the formula for the nth term: \(a_5 = 1 \cdot \left(-\frac{x}{3}\right)^{5-1} = \left(-\frac{x}{3}\right)^4\). Calculate \((-1)^4 = 1\) and \(\left(\frac{x}{3}\right)^4 = \frac{x^4}{81}\), so \(a_5 = \frac{x^4}{81}\).

Step 4: Find the Eighth Term

Replace \(n\) with 8 in the formula for the nth term: \(a_8 = 1 \cdot \left(-\frac{x}{3}\right)^{8-1} = \left(-\frac{x}{3}\right)^7\). Calculate \((-1)^7 = -1\) and \(\left(\frac{x}{3}\right)^7 = \frac{x^7}{2187}\), so \(a_8 = -\frac{x^7}{2187}\).

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

One of the foundational concepts in geometric sequences is the common ratio. This ratio is crucial because it determines how each term in the sequence is generated from the previous term. To find the common ratio, you simply divide any term in the sequence by the term that comes directly before it. In our specific exercise, the first term of the sequence is given as 1, and the second term is \(-\frac{x}{3}\).

By dividing the second term by the first term, the common ratio \(r\) is determined as \(-\frac{x}{3}\). The common ratio remains constant throughout the sequence, constantly multiplying the previous term to arrive at the next. Here's why it's significant:

• It tells you the pattern or multiplier of growth or decay in the sequence.
• Knowing the common ratio allows you to easily compute other terms in the sequence using the formula for the nth term.

Make sure to accurately calculate it to avoid errors in finding subsequent terms.

Using the nth Term Formula to Calculate Any Term

To understand geometric sequences better, you'll need to get familiar with the nth term formula. This handy formula allows you to find any term in the sequence if you know the first term and the common ratio.

The formula is \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio, and \(n\) is the number in the sequence.

In our example:

• First term \(a_1 = 1\)
• Common ratio \(r = -\frac{x}{3}\)

Plug these into the formula for any \(n\)th term and solve. For instance, filling in \(n = 5\) gives you the fifth term \(a_5 = \left(-\frac{x}{3}\right)^4 = \frac{x^4}{81}\).

This formula is versatile and very useful, making it possible to compute any term without having to manually calculate each preceding term. It's like having a map for your journey through the sequence.

Identifying the First Term of a Geometric Sequence

The first term in a geometric sequence is where everything starts. It is the initial number from which you multiply by the common ratio to generate the rest of the sequence. Knowing the first term is crucial because it forms the base of your calculations for future terms.

In the given sequence, the first term is provided as 1.

• This is your starting point for using the nth term formula.
• It’s essential for maintaining accuracy when finding terms further down the sequence.

Without the first term, you would lack a reference point, making it impossible to implement the nth term formula accurately. Think of it as the foundation upon which you build your entire sequence.