Problem 27 - Algebra and Trigonometry

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

Common ratio: \(-1/12\), fifth term: \(1/144\), \(n\)th term: \(144 \cdot (-1/12)^{(n-1)}\).

Step 1: Identify the Common Ratio

A geometric sequence has a constant ratio between successive terms. To find the common ratio \(r\), divide the second term by the first term: \(-12 / 144 = -1/12\). Check by dividing any other successive terms: \(1 / -12 = -1/12\) and \(-1/12 / 1 = -1/12\). So, the common ratio is \(r = -\frac{1}{12}\).

Step 2: Calculate the Fifth Term

To find the fifth term, 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 \(n = 5\). Substitute these values: \[a_5 = 144 \cdot \left(-\frac{1}{12}\right)^{4} = 144 \cdot \left(\frac{1}{20736}\right) = \frac{144}{20736} = \frac{1}{144}.\] So, the fifth term is \(\frac{1}{144}\).

Step 3: Derive the Formula for the nth Term

Given that \(a_1 = 144\) and \(r = -\frac{1}{12}\), the formula for the \(n\)th term \(a_n\) of a geometric sequence is \[a_n = a_1 \cdot r^{(n-1)} = 144 \cdot \left(-\frac{1}{12}\right)^{(n-1)}.\].

Key Concepts

Common Ratio

In a geometric sequence, discovering the common ratio, denoted as \(r\), is a crucial step. The common ratio is what distinguishes a geometric sequence from other types of sequences. It is the factor by which we multiply each term to get the next term.
For example, in the sequence \(144, -12, 1, -\frac{1}{12}, \ldots\), to find \(r\), we divide the second term by the first term:

• \(-12 \div 144 = -\frac{1}{12}\)
• To ensure consistency, check with another pair: \(1 \div (-12) = -\frac{1}{12}\)
• Lastly, \(-\frac{1}{12} \div 1 = -\frac{1}{12}\)

This confirms our common ratio \(r = -\frac{1}{12}\). Noticing that the ratio remains constant reassures we're dealing with a genuine geometric sequence.

Nth Term

The \(n\)th term of a geometric sequence provides a way to find any term in the sequence without listing all previous terms. The explicitly defined formula for this term is \(a_n = a_1 \cdot r^{(n-1)}\), where:

• \(a_1\) is the first term, and
• \(r\) is the common ratio.
• \(n\) is the term number we wish to find.

To find the fifth term, or \(a_5\), for example, you just substitute:

• \(a_1 = 144\),
• \(r = -\frac{1}{12}\), and
• \(n = 5\)

This leads to \(a_5 = 144 \cdot \left(-\frac{1}{12}\right)^4\), which simplifies to \(\frac{1}{144}\). This approach allows us to quickly determine any term irrespective of the sequence's length.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. Each term is multiplied by the common ratio to obtain the next term in the series. If you have a finite geometric series, you can find the sum \(S_n\). The formula is:

• \(S_n = a_1 \frac{1 - r^n}{1 - r}\), where \(r eq 1\).

Suppose we need to sum the first five terms of our sequence \(144, -12, 1, -\frac{1}{12}, \frac{1}{144}\). We can calculate:

• \(S_5 = 144 \frac{1 - \left(-\frac{1}{12}\right)^5}{1 - (-\frac{1}{12})}\)

This series sum provides an insightful way to view the cumulative effect of terms in a sequence.

Sequence Formula

In the realm of sequences, the sequence formula serves as a map to navigate through the terms of a sequence. For geometric sequences, it's all about leveraging the formula for the \(n\)th term: \(a_n = a_1 \cdot r^{(n-1)}\).
This formula helps us to:

• Predict future terms without manual calculation.
• Understand the behavior of sequence over time.

For our sequence, knowing that \(a_1 = 144\) and \(r = -\frac{1}{12}\) gives a precise formula. You just substitute any term number \(n\) into \(a_n = 144 \cdot \left(-\frac{1}{12}\right)^{(n-1)}\) to find the desired term. By using this sequence formula, comprehending the propagation of values throughout the sequence becomes simple and straightforward.