Problem 34 - 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

The common ratio is \(\frac{1}{4}\), the fifth term is \(-\frac{1}{32}\), and the nth term is \(-8 \cdot \left(\frac{1}{4}\right)^{n-1}\).

Step 1: Identify the Sequence Elements

The given geometric sequence is \(-8, -2, -\frac{1}{2}, -\frac{1}{8}, \dots\). We identify that these terms are the first four terms of the sequence.

Step 2: Determine the Common Ratio

A geometric sequence has each term as the previous term multiplied by a common ratio \(r\). To find \(r\), divide the second term by the first term: \( r = \frac{-2}{-8} = \frac{1}{4} \). Let's check consistency: \( \frac{-\frac{1}{2}}{-2} = \frac{1}{4} \), \( \frac{-\frac{1}{8}}{-\frac{1}{2}} = \frac{1}{4} \). The common ratio \(r\) is consistently \( \frac{1}{4} \).

Step 3: Calculate the Fifth Term

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

Step 4: Derive the General Formula for the nth Term

Using the same formula for the \(n\)th term: \( a_n = a_1 \cdot r^{n-1} \). Substitute \(a_1 = -8\) and \(r = \frac{1}{4}\) to get \[ a_n = -8 \cdot \left(\frac{1}{4}\right)^{n-1}. \] This is the formula to find any term in the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the key feature is that each term is derived from the previous one by multiplying a constant, known as the 'common ratio'. This value is consistent throughout the sequence, and determining it is crucial for analyzing and predicting further terms of the sequence.
To find the common ratio, take any term in the sequence and divide it by the term that comes before it. For our given example \(-8, -2, -\frac{1}{2}, -\frac{1}{8}, \dots\),the common ratio \(r\) is calculated by dividing the second term by the first term: \[r = \frac{-2}{-8} = \frac{1}{4}.\]Verifying this with other terms ensures the consistency of this common ratio. Thus, the common ratio in our sequence is consistently \(\frac{1}{4}\).
Understanding this ratio helps us establish the multiplication factor needed to find successive terms.

nth Term of a Sequence

The 'nth term' of a sequence refers to any term located at position \(n\) in a series of numbers. It's a general form used to find any term in a sequence without explicitly listing all preceding terms.
In geometric sequences like ours, the formula to calculate the \(n\)th term, often represented as \(a_n\), is specified by\[a_n = a_1 \cdot r^{n-1},\]where \(a_1\) is the first term, and \(r\) is the common ratio. This formula captures the exponential nature of geometric sequences and helps us calculate terms effectively.
For instance, in our sequence, with \(a_1 = -8\) and \(r = \frac{1}{4}\), to determine the fifth term, substitute these values into the formula:\[a_5 = -8 \cdot \left(\frac{1}{4}\right)^4 = -\frac{1}{32}.\]

Sequence Formula

The sequence formula provides a mathematical expression to find any term of the sequence. For geometric sequences, this involves the initial term and the common ratio. The general formula, \[a_n = a_1 \cdot r^{n-1},\]is tailored for each specific sequence by plugging in the value of the first term and the identified common ratio.
In our case, for the sequence \(-8, -2, -\frac{1}{2}, -\frac{1}{8}, \dots\),the first term \(a_1\) is \(-8\),and the ratio \(r\) is\(\frac{1}{4}\). By substituting these values into the formula, we derive:\[a_n = -8 \cdot \left(\frac{1}{4}\right)^{n-1}.\]This expression gives us the power to find any term without reconstructing the entire sequence.
Using this formula can simplify many problems and allow deeper exploration of sequence patterns.