Problem 25 - College Algebra

Question

Write an explicit formula for each sequence. $$1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \frac{1}{16}, \dots$$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = \left(-\frac{1}{2}\right)^{n-1} \).

Step 1: Identify the First Term

The first step is to identify the first term in the sequence. Here, the first term is given as \( a_1 = 1 \).

Step 2: Recognize the Pattern

Look at the sequence to find the pattern. This sequence alternates in sign and halves each previous term. The denominators can be expressed as powers of 2: 2, 4, 8, 16, etc., which corresponds to \(2^1, 2^2, 2^3, 2^4, \) and so on.

Step 3: Determine if It's Arithmetic or Geometric

Check if the sequence is arithmetic or geometric. An arithmetic sequence changes by addition or subtraction, while a geometric sequence multiplies by a fixed number. This sequence multiplies by \(-\frac{1}{2}\) with each subsequent term, suggesting it's geometric.

Step 4: Write the General Form of a Geometric Sequence

The general formula for the \( n \)-th term of a geometric sequence is given by \( a_n = a_1 \cdot r^{n-1} \), where \( r \) is the common ratio. Here, \( a_1 = 1 \) and the common ratio \( r = -\frac{1}{2} \).

Step 5: Formulate the Explicit Formula

Substitute \( a_1 = 1 \) and \( r = -\frac{1}{2} \) into the general formula. Thus, the explicit formula for the sequence is \( a_n = 1 \cdot \left(-\frac{1}{2}\right)^{n-1} \) or simply \( a_n = \left(-\frac{1}{2}\right)^{n-1} \).

Key Concepts

Explicit Formula

The explicit formula is a powerful tool that allows us to calculate any term in a sequence without needing to know the previous terms. It provides a direct relationship between the position of a term in the sequence and its value. For a geometric sequence, the explicit formula is given by \( a_n = a_1 \cdot r^{n-1} \). This means that to find any term \( a_n \), you multiply the first term \( a_1 \) by the common ratio \( r \) raised to the power of \( (n-1) \), where \( n \) is the term number. In the sequence 1, \(-\frac{1}{2}\), \(\frac{1}{4}\), \(-\frac{1}{8}\), \(\frac{1}{16}\),\( \ldots \), the explicit formula is \( a_n = \left(-\frac{1}{2}\right)^{n-1} \). This formula works for any term of this sequence: it gives negative terms for odd \( n \) and positive terms for even \( n \), thanks to the alternation in signs caused by the negative common ratio. Try using different values of \( n \) to see how each term is calculated directly from this formula.

Common Ratio

The common ratio is the number that each term of a geometric sequence is multiplied by to get the next term. Understanding this ratio is crucial because it defines a geometric sequence and allows us to write the explicit formula. In our sequence 1, \(-\frac{1}{2}\), \(\frac{1}{4}\), \(-\frac{1}{8}\), \(\frac{1}{16}\), \( \ldots \), the common ratio \( r \) is \(-\frac{1}{2}\). This means that each term is \(-\frac{1}{2}\) times the preceding term.

• The common ratio determines the direction (increase or decrease) and behavior (growth, decay, oscillation) of the sequence.
• If the absolute value of \( r \) is less than 1, like in this example, the sequence values get smaller. When the absolute value is greater than 1, the terms grow larger.

In this particular sequence, as you can see, every second term changes sign, causing an alternating pattern of positive and negative values. The sign change is a direct result of multiplying by the negative common ratio.

Sequence Patterns

Recognizing patterns in a sequence is essential to understanding and classifying that sequence. Patterns can help us determine if a sequence is arithmetic, geometric, or something else entirely. In a geometric sequence, the pattern is determined by consistent multiplication (or division) by a common ratio. When you start to evaluate the sequence 1, \(-\frac{1}{2}\), \(\frac{1}{4}\), \(-\frac{1}{8}\), \(\frac{1}{16}\), \( \ldots \), you can quickly spot the pattern by observing the calculation: each term is halving the magnitude of the previous and switching signs due to multiplication by \(-\frac{1}{2}\).

• The pattern is critical in ensuring that the sequence remains geometric, with no irregularities breaking the flow.
• Identifying the rate of change helps confirm if the pattern fits the geometric sequence category.

By observing and understanding the structure of the sequence, it becomes easier to predict future terms and to write a direct formula that provides these terms.