Problem 27 - Precalculus: Mathematical 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

Common ratio: \(-\frac{1}{12}\), fifth term: \(\frac{1}{144}\), nth term: \(a_n = 144 \cdot \left(-\frac{1}{12}\right)^{(n-1)}\).

Step 1: Identify the First Term

Find the first term, which is the starting point of the sequence. The first term of the sequence is 144.

Step 2: Determine the Common Ratio

To find the common ratio, divide the second term by the first term: \(r = \frac{-12}{144} = -\frac{1}{12}\). The common ratio is \(-\frac{1}{12}\).

Step 3: Calculate the Fifth Term

Use the formula for the nth term of a geometric sequence: \(a_n = a_1 \cdot r^{(n-1)}\). Substitute \(n=5\), \(a_1=144\), and \(r=-\frac{1}{12}\) into the formula: \[ a_5 = 144 \cdot \left(-\frac{1}{12}\right)^{(5-1)} = 144 \cdot \left(-\frac{1}{12}\right)^4 \]. Calculate \(\left(-\frac{1}{12}\right)^4 = \frac{1}{20736}\), so \(a_5 = 144 \cdot \frac{1}{20736} = \frac{1}{144}\). The fifth term is \(\frac{1}{144}\).

Step 4: Write the Formula for the nth Term

Use the formula for the nth term again: \(a_n = a_1 \cdot r^{(n-1)}\). With \(a_1 = 144\) and \(r = -\frac{1}{12}\), the nth term formula is \(a_n = 144 \cdot \left(-\frac{1}{12}\right)^{(n-1)}\). This is the expression for the nth term of the sequence.

Key Concepts

Common Ratio

In the world of geometric sequences, the common ratio is a crucial element. It is the factor by which we multiply one term to get the next term in the sequence. Identifying the common ratio is essential for understanding the behavior of the sequence over time.
To find the common ratio, simply divide any term in the sequence by the term that comes before it. For example, if you have the sequence: 144, -12, 1, \(-\frac{1}{12}\),...

• The first term (\(a_1\)) is 144.
• The second term is -12.

To find the common ratio \(r\), perform the following division: \(r = \frac{-12}{144} = -\frac{1}{12}\).
This tells us that every term is multiplied by \(-\frac{1}{12}\) to get to the next term.

Nth Term Formula

The nth term formula in a geometric sequence is a generalized formula that enables us to find any term in the sequence without listing out all the preceding terms. This formula is extremely useful for large sequences where writing out each term would be impractical.
In a geometric sequence, the nth term, denoted as \(a_n\), can be calculated using the formula: \[ a_n = a_1 \cdot r^{(n-1)} \] - \(a_1\) is the first term of the sequence.- \(r\) is the common ratio.- \(n\) represents the term number you want to find.
Using the earlier example where \(a_1 = 144\) and \(r = -\frac{1}{12}\), the formula becomes: \[ a_n = 144 \cdot \left(-\frac{1}{12}\right)^{(n-1)} \]
Plug in any positive integer for \(n\) to find the respective term. This shows how the sequence progresses or expands over time.

Fifth Term Calculation

Calculating specific terms in a geometric sequence, like the fifth term, involves substituting values into the nth term formula. Let's see how this is done with our sequence having first term \(a_1 = 144\) and common ratio \(r = -\frac{1}{12}\).
To find the fifth term \(a_5\) of our sequence, we set \(n=5\) in the nth term formula: \[ a_5 = 144 \cdot \left(-\frac{1}{12}\right)^{(5-1)} \] Now, calculate the power: \( \left(-\frac{1}{12}\right)^4 = \frac{1}{20736} \) Substitute this back into the equation: \( a_5 = 144 \cdot \frac{1}{20736} \) This simplifies to \( a_5 = \frac{1}{144} \).
This outcome illustrates how the sequence behaves at the fifth position, giving us insight into both the smaller scale (individual terms) and the broader pattern of the sequence.