Problem 31 - Precalculus Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$144,-12,1,-\frac{1}{12}, \dots$$

Step-by-Step Solution

Answer

The common ratio is \(-\frac{1}{12}\), the fifth term is \(\frac{1}{144}\), and the nth term is \(a_n = 144 \times \left(-\frac{1}{12}\right)^{n-1}\).

Step 1: Identify the Common Ratio

A geometric sequence progresses by a constant factor called the common ratio. To find the common ratio, divide any term by the previous term. Let's calculate the common ratio using the first two terms: \(-12 / 144\). This gives us the common ratio \(r = -\frac{1}{12}\).

Step 2: Calculate the Fifth Term

We use the formula for the nth term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\). To find the fifth term, substitute \(a_1 = 144\), \(r = -\frac{1}{12}\), and \(n = 5\): \(a_5 = 144 \times \left(-\frac{1}{12}\right)^{4}\). Simplifying, we have \(a_5 = 144 \times \left(\frac{1}{20736}\right) = \frac{1}{144}\).

Step 3: Derive the General Formula for the Nth Term

The nth term of a geometric sequence is calculated by the formula \(a_n = a_1 \times r^{n-1}\). For this sequence, substitute \(a_1 = 144\) and \(r = -\frac{1}{12}\). Therefore, the nth term is \(a_n = 144 \times \left(-\frac{1}{12}\right)^{n-1}\).

Key Concepts

Common Ratio

The common ratio in a geometric sequence is a crucial element that defines the sequence's progression. It is the constant factor by which each term is multiplied to get the next term. Finding the common ratio is simple: you divide one term by the previous one.
For example, in the sequence given in the exercise, where the first term is 144 and the second term is -12, the common ratio is calculated as follows: \[ r = \frac{-12}{144} = -\frac{1}{12} \]This value of \(-\frac{1}{12}\) remains the same throughout the sequence. It means each term is obtained by multiplying the previous term by \(-\frac{1}{12}\). A negative common ratio results in terms that alternate in sign, as seen in this sequence.

Nth Term Formula

The nth term formula of a geometric sequence helps in finding any term in the sequence. It's a potent tool because it allows you to calculate any term without listing all of the previous ones. The formula is:\[ a_n = a_1 \times r^{n-1} \] Where:

• \(a_n\) is the term you're solving for.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number.

This formula simplifies the process by providing a direct route to any term in the sequence, rather than needing to multiply repeatedly. For instance, when looking for the fifth term in the exercise, using the formula gives us:\[ a_5 = 144 \times \left(-\frac{1}{12}\right)^{4} = \frac{1}{144} \] This ensures accuracy and saves time by skipping multiple steps.

Geometric Progression

Geometric progression is all about a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Characteristics of a geometric progression include:

• Consistent multiplication pattern.
• Terms either steadily increase, decrease, or alternate in sign.
• Easy to predict future terms using the nth term formula.

In the example from the exercise, the sequence \(144, -12, 1, -\frac{1}{12}, \ldots\) is a geometric progression because each term is obtained by multiplying the previous term by \(-\frac{1}{12}\). Understanding geometric progressions is key in various mathematical and real-world applications, such as calculating interest, analyzing population growth, and much more. Recognizing a sequence as geometric allows you to apply specialized formulas and insights specific to this type of sequence.