Problem 31 - Beginning and Intermediate Algebra

Question

Find a formula for the general term, \(a_{n},\) of each sequence. $$-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \frac{1}{16}, \dots$$

Step-by-Step Solution

Answer

The general term formula for the given sequence is: \[ a_n = (-1)^n \cdot (\frac{1}{2})^n \]

Step 1: Identify the signs pattern

We can see that in the given sequence, the signs for each term alternate between negative and positive. For the general term formula, we need to represent this alternating pattern. We can use the term \((-1)^n\) for this purpose, where n is the position of the term in the sequence.

Step 2: Identify the absolute value pattern

In the sequence, the absolute value of each term is halved to obtain the absolute value of the next term. Given that the first term's absolute value is 1/2, we can represent this pattern as \( (\frac{1}{2})^n \) for the general term formula, where n is the position of the term in the sequence.

Step 3: Combine the patterns

We combine the patterns for both the sign and the absolute value to form the general term formula. Since we need to multiply both patterns, we have: \[ a_n = (-1)^n \cdot (\frac{1}{2})^n \] However, this general term formula assumes that the first term of the sequence corresponds to n=1. Since we found the correct pattern for the entire sequence, this is our general term formula for the given sequence. The general term formula for the given sequence is: \[ a_n = (-1)^n \cdot (\frac{1}{2})^n \]

Key Concepts

Alternating Signs in Sequences

In sequences, alternating signs refer to terms changing between positive and negative. This pattern is crucial when determining the sequence behavior. For this sequence:

• Each term switches signs, leading to a consistent pattern.
• Alternating signs are represented mathematically using the expression \((-1)^n\). This expression results in odd terms being negative, while even terms are positive.

In our case, starting from the first term which is negative, this sign change effectively emphasizes the sequence's oscillating nature. As they switch between negative and positive, knowing this pattern helps us in formulating the general term for a sequence.

General Term Formula of a Sequence

Creating a general term formula is essentially finding an expression that allows you to compute any term in the sequence without listing them all. This means forming an expression in terms of \( n \), which denotes the position of a term in the sequence.

• Think of the formula as a rule that can generate the nth term.
• In the given sequence, we see a clear pattern of signs and decreasing magnitude.

Thus, we discern the formula: \[ a_n = (-1)^n \cdot \left(\frac{1}{2}\right)^n \]. This equation combines both the sign pattern and the magnitude pattern to provide a concise form for all terms. Knowing how to construct such formulas allows deeper insight into the behavior of complex sequences.

Understanding the Absolute Value Pattern

The absolute value pattern in a sequence shows how the magnitude of each term changes irrespective of its sign. Observing this can help detect underlying geometric patterns.

• Here, the absolute value decreases by half as we move to each subsequent term.
• Mathematically, this diminishing pattern is represented by \( \left(\frac{1}{2}\right)^n \), signifying an exponential decay with respect to the term's position.

This idea of an absolute value pattern is essential for constructing the magnitude part of the general term formula, helping us predict future terms' magnitude based on their position in the sequence.