Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each sequence. \(1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \frac{1}{16}, \ldots\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern

Examine the sequence: \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, \ldots\). Notice that the signs alternate between positive and negative, and the denominators indicate powers of 2.

Step 2: Determine the Common Ratio

Identify the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{-\frac{1}{2}}{1} = -\frac{1}{2}\). Verify by checking subsequent terms: \(\frac{\frac{1}{4}}{-\frac{1}{2}} = -\frac{1}{2}\), confirming the common ratio is \(-\frac{1}{2}\).

Step 3: Write the General Form of the Sequence

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

Step 4: Construct the Explicit Formula

Using the identified first term and common ratio, the explicit formula for the sequence is: \(a_n = 1 \cdot \left(-\frac{1}{2}\right)^{n-1}\). Thus, \(a_n = \left(-\frac{1}{2}\right)^{n-1}\).

Key Concepts

Common Ratio

In a geometric sequence, understanding the _common ratio_ is key to grasping its underlying structure. The common ratio is the factor by which each term of the sequence is multiplied to get the next term. It remains constant across the sequence.
For instance, in the sequence provided: \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, \ldots\), to find the common ratio, you can simply divide the second term by the first:

• \( r = \frac{-\frac{1}{2}}{1} = -\frac{1}{2} \)

You can also confirm the ratio remains consistent by checking subsequent terms:

• \( r = \frac{\frac{1}{4}}{-\frac{1}{2}} = -\frac{1}{2} \)

This shows that the sequence is consistently multiplied by \(-\frac{1}{2}\). Recognizing this pattern is essential, as it helps in both predicting future terms in the sequence and constructing formulas like the explicit formula for the sequence.

Explicit Formula

The _explicit formula_ for a geometric sequence provides a way to calculate any term without having to know all the previous terms. It saves you from endless calculations and gives a direct jump from the position in the sequence to the value of that term.
The explicit formula for a geometric sequence is expressed as:

• \( a_n = a_1 \cdot r^{n-1} \)

Here, \(a_1\) represents the first term of the sequence, \(n\) is the position of the term within the sequence, and \(r\) is the common ratio.
For the sequence \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, \ldots\), the first term \(a_1\) is \(1\), and we've calculated the common ratio \(r\) as \(-\frac{1}{2}\).
Thus, the explicit formula becomes:

• \( a_n = 1 \cdot \left(-\frac{1}{2}\right)^{n-1} \)

So, for any term \(n\), you just substitute \(n\) into this formula to find the term's value readily.

Alternating Series

In examining sequences, the term _alternating series_ describes a series of numbers where the sign of each term alternates between positive and negative. This property is seen in various mathematical applications and can be deduced by observing the signs of consecutive terms.
Consider the sequence \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, \ldots\). If you look closely, the signs alternate for each successive term:

• Positive: \(1\), \(\frac{1}{4}\), \(\frac{1}{16}\), ...
• Negative: \(-\frac{1}{2}\), \(-\frac{1}{8}\), ...

Such alternating patterns are very characteristic of geometric sequences where the common ratio is negative. The negative ratio ensures that multiplying terms by it switches their signs, perfectly matching the pattern of alternating series.
Understanding these sign changes is crucial for both constructing and interpreting the sequence, as it influences calculations and insights regarding the sequence's behavior and sum.