Problem 33 - 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 \(n\) th term is \(144 \cdot \left(-\frac{1}{12}\right)^{n-1}\)."

Step 1: Identify Two Consecutive Terms

Begin by identifying any two consecutive terms in the sequence. For this sequence, let's pick the first term 144 and the second term -12.

Step 2: Calculate the Common Ratio

The common ratio \( r \) of a geometric sequence can be calculated by dividing the second term by the first term.\[r = \frac{-12}{144} = -\frac{1}{12}\]

Step 3: Determine the Fifth Term

To find the fifth term of the sequence, use the formula for the \( n \) th term of a geometric sequence: \( a_n = a_1 \cdot r^{n-1} \). Here, \( a_1 = 144 \), \( r = -\frac{1}{12} \), and \( n = 5 \).\[a_5 = 144 \cdot \left(-\frac{1}{12}\right)^{4} = 144 \cdot \frac{1}{20736} = \frac{1}{144}\]

Step 4: General Formula for the n-th Term

The formula for the \( n \) th term is \( a_n = a_1 \cdot r^{n-1} \). By plugging in the known values of \( a_1 \) and \( r \), we get\[a_n = 144 \cdot \left(-\frac{1}{12}\right)^{n-1}\]

Key Concepts

Common Ratio

In geometric sequences, the common ratio is a crucial concept. It helps us understand how the sequence behaves from one term to the next. The common ratio, usually denoted as \( r \), is found by dividing one term in the sequence by the previous term. When you look at the sequence \(144, -12, 1, -\frac{1}{12}, \dots\), you'll notice each term is obtained by multiplying the previous term by the common ratio.

For example, to find the common ratio of our sequence, take \(-12\) (the second term) and divide it by \(144\) (the first term):

• \( r = \frac{-12}{144} = -\frac{1}{12} \)

This tells us that each term is \(-\frac{1}{12}\) of the previous term. Understanding the common ratio gives you the power to predict any term in the sequence.

n-th Term Formula

The \( n \)-th term formula is an essential tool in analyzing geometric sequences. It lets you find any term in the sequence without listing all previous terms. This formula is expressed as:

• \( a_n = a_1 \cdot r^{n-1} \)

Here, \( a_n \) is the \( n \)-th term you want to find, \( a_1 \) is the first term of the sequence, and \( r \) is the common ratio. The exponent \( n-1 \) represents the number of steps you take from the first term to the \( n \)-th term.

In the given sequence, the first term \( a_1 = 144 \) and the common ratio \( r = -\frac{1}{12}\). To find the general term, substitute these values into the formula:

• \( a_n = 144 \cdot \left(-\frac{1}{12}\right)^{n-1} \)

This formula is a powerful tool because it lets you calculate any term in the sequence quickly and easily.

Fifth Term of a Sequence

Once you have the n-th term formula, calculating specific terms becomes straightforward. For instance, if you want to find the fifth term of the geometric sequence, you substitute \( n = 5 \) into the formula:

• \( a_5 = 144 \cdot \left(-\frac{1}{12}\right)^{4} \)

Let's break down the calculations:

• First, calculate \( \left(-\frac{1}{12}\right)^{4} \). This causes the negative sign to disappear because of the even exponent, resulting in a positive fraction:
• \( \left(-\frac{1}{12}\right)^{4} = \frac{1}{20736} \).
• Multiply this by the first term (144): \( a_5 = 144 \cdot \frac{1}{20736} \).

So, the fifth term, \( a_5 \), equals \( \frac{1}{144} \). Understanding how to use the n-th term formula in this way makes it easy to pinpoint any term in the sequence, even if it has many terms.