Problem 22 - Contemporary Precalculus

Question

Find a formula for the nth term of the sequence whose first few terms are given. $$-\frac{1}{8},-\frac{1}{2},-2,-8,-32, \dots$$

Step-by-Step Solution

Answer

Answer: The formula for the nth term of the given sequence is \(a_n = -\frac{1}{8} \cdot 4^{(n-1)}\).

Step 1: Identify the first term

The first term in the sequence is given as \(-\frac{1}{8}\). That is denoted by \(a_1\).

Step 2: Calculate the common ratio

To find the common ratio, divide any term in the sequence by its previous term. For example, dividing the second term by the first term, we have \(r = \frac{-\frac{1}{2}}{-\frac{1}{8}}\). Calculate this ratio: $$r = \frac{\frac{1}{2}}{\frac{1}{8}} = \frac{1}{2} \times \frac{8}{1} = 4$$

Step 3: Find the general formula for the nth term

The general formula for the nth term of a geometric sequence is given by \(a_n = a_1 \times r^{(n-1)}\). Use the values obtained in steps 1 and 2, we have \(a_n = -\frac{1}{8} \times 4^{(n-1)}\). So, the formula for the nth term of the given sequence is: $$a_n = -\frac{1}{8} \cdot 4^{(n-1)}$$.

Key Concepts

nth term formula

When examining a geometric sequence, one key task is finding the formula for the nth term. This formula allows us to describe the sequence's behavior mathematically. A geometric sequence is defined by its first term, denoted as \(a_1\), and its common ratio, \(r\).
In general, the nth term \(a_n\) of a geometric sequence can be calculated using the formula: \[ a_n = a_1 \times r^{(n-1)} \] Here, \(a_n\) represents the nth term you want to find, and \(n\) is the position of that term within the sequence. This formula is powerful because it enables the calculation of any term in the sequence without having to list all preceding terms.
For example, using our sequence with \(a_1 = -\frac{1}{8}\) and \(r = 4\), the nth term is given by \[ a_n = -\frac{1}{8} \times 4^{(n-1)} \] Applying this makes computation more efficient, especially for large \(n\), providing a direct method to reach any term in the sequence.

common ratio

The common ratio is a crucial element in the analysis of a geometric sequence. It indicates how each term relates to the one before it. To find the common ratio \(r\), divide any term by the term preceding it. This ratio stays consistent throughout the entire sequence.

• For our sequence: \(-\frac{1}{8}, -\frac{1}{2}, -2,\) and so on, divide \(-\frac{1}{2}\) by \(-\frac{1}{8}\)
• The calculation goes as follows: \[ r = \frac{-\frac{1}{2}}{-\frac{1}{8}} = \frac{\frac{1}{2}}{\frac{1}{8}} = \frac{1}{2} \times \frac{8}{1} = 4 \]

Here, \(r = 4\), telling us each term is four times its predecessor. Recognizing the constant ratio is essential, as it governs the sequence's multiplication pattern and ensures the correct application of the nth term formula. Without a constant common ratio, a sequence ceases to be geometric.

sequence analysis

Sequence analysis in the context of geometry involves examining the pattern and behavior of numbers within a sequence. For geometric sequences, this means understanding how each term scales from the previous one via the common ratio. This insight leads us to appreciate the inherent simplicity and consistency within these sequences.
When analyzing a geometric sequence like \(-\frac{1}{8}, -\frac{1}{2}, -2, -8, -32, \dots\), start by identifying basic characteristics:

• Initial term \(a_1\): The very first number in the sequence.
• Common ratio \(r\): A factor that determines how each subsequent term evolves from its predecessor.

By substituting into the nth term formula, the sequence can be fully described. Thus unfolds each subsequent term's value without manual multiplication for every step.
This method of analysis aids in predicting continued growth patterns and understanding the underlying mathematical structure, thus offering both simplicity and efficiency in handling sizable sequences.