Problem 32 - Precalculus Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$-8,-2,-\frac{1}{2},-\frac{1}{8}, \dots$$

Step-by-Step Solution

Answer

Common ratio is \( \frac{1}{4} \), fifth term is \(-\frac{1}{32}\), \( n \)th term is \( -8 \left(\frac{1}{4}\right)^{(n-1)} \)."

Step 1: Determine the Common Ratio

To find the common ratio in a geometric sequence, divide any term by the previous term. Let's use the first two terms: \[ r = \frac{-2}{-8} = \frac{1}{4} \] The common ratio \( r \) is \( \frac{1}{4} \).

Step 2: Find the Fifth Term

To find the fifth term, use the formula for the \( n \) th term of a geometric sequence, which is \( a_n = a_1 \cdot r^{(n-1)} \). Here, \( a_1 = -8 \), \( r = \frac{1}{4} \), and \( n = 5 \). \[ a_5 = -8 \cdot \left(\frac{1}{4}\right)^{4} \] Calculate:\[ a_5 = -8 \cdot \frac{1}{256} = -\frac{8}{256} = -\frac{1}{32} \] The fifth term is \(-\frac{1}{32}\).

Step 3: General Formula for the \(n\)th Term

Using the formula for the \( n \) th term \( a_n = a_1 \cdot r^{(n-1)} \), substituting \( a_1 = -8 \) and \( r = \frac{1}{4} \):\[ a_n = -8 \cdot \left(\frac{1}{4}\right)^{(n-1)} \] This is the expression for the \( n \) th term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a fundamental element that helps us understand how each term relates to the previous one. It is the factor by which we multiply a term to get to the next term in the sequence. In this particular problem, we determine the common ratio by dividing any term by the preceding term.

Consider the initial terms of our sequence:

• First term: \(-8\)
• Second term: \(-2\)

To find the common ratio \(r\), we divide the second term by the first term:\[ r = \frac{-2}{-8} = \frac{1}{4}\]Thus, the common ratio \(r\) is \(\frac{1}{4}\).
This value is crucial as it denotes the constant multiplier connecting each term of the sequence.

Nth Term Formula

The \( n \)th term formula in a geometric sequence helps predict any term's value based on its position within the sequence. This formula is particularly useful for sequences with a constant ratio, as it allows calculating the value without listing all preceding terms.

The formula is given by:\[a_n = a_1 \cdot r^{(n-1)}\]

• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number we are interested in.

For our sequence, substitute \( a_1 = -8 \) and \( r = \frac{1}{4} \) into the formula:\[a_n = -8 \cdot \left(\frac{1}{4}\right)^{(n-1)}\]This signifies that to obtain the \( n \)th term, we multiply the first term by the common ratio raised to the power of \( n-1 \).
This formula provides a straightforward methodology to calculate any term in the sequence, thereby ensuring quick and accurate results.

Fifth Term Calculation

Calculating the fifth term of a geometric sequence involves applying the \( n \)th term formula specifically, while substituting the corresponding term number (in this case, 5). This procedure exemplifies the practical use of the \( n \)th term formula.

For the fifth term, set \( n = 5 \). Using the given values:

• First term, \( a_1 = -8 \)
• Common ratio, \( r = \frac{1}{4} \)

Substitute these into the formula:\[a_5 = -8 \cdot \left(\frac{1}{4}\right)^4\]Simplify the result:\[a_5 = -8 \cdot \frac{1}{256} = -\frac{1}{32}\]Therefore, the fifth term of the sequence is \(-\frac{1}{32}\).
This result showcases how the \( n \)th term formula enables us to efficiently find specific terms without listing the entire sequence.