Problem 11 - College Algebra

Question

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. $$ -1, \frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \ldots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \(-\frac{1}{2}\).

Step 1: Identify the Pattern in the Sequence

First, write down the sequence to observe the terms: \(-1, \frac{1}{2}, -\frac{1}{4}, \frac{1}{8}, -\frac{1}{16}, \ldots\). Notice the alternating signs and decreasing magnitudes of the fractions.

Step 2: Calculate Successive Ratios

To determine if the sequence is geometric, calculate the ratio between successive terms:\[ r_1 = \frac{\frac{1}{2}}{-1} = -\frac{1}{2} \]\[ r_2 = \frac{-\frac{1}{4}}{\frac{1}{2}} = -\frac{1}{2} \] \[ r_3 = \frac{\frac{1}{8}}{-\frac{1}{4}} = -\frac{1}{2} \] \[ r_4 = \frac{-\frac{1}{16}}{\frac{1}{8}} = -\frac{1}{2} \] Each ratio calculated is \(-\frac{1}{2}\).

Step 3: Conclude the Pattern

Since the ratio between successive terms is constant at \(-\frac{1}{2}\), the sequence is geometric.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The common ratio (\( r \)) is what distinguishes a geometric sequence from others, like arithmetic sequences. To find the common ratio, simply divide any term in the sequence by the previous term.
For example, in the sequence \(-1, \frac{1}{2}, -\frac{1}{4}, \frac{1}{8}, -\frac{1}{16}, \ldots \), we identified the pattern by computing successive ratios:

• \( r_1 = \frac{\frac{1}{2}}{-1} = -\frac{1}{2} \)
• \( r_2 = \frac{-\frac{1}{4}}{\frac{1}{2}} = -\frac{1}{2} \)
• \( r_3 = \frac{\frac{1}{8}}{-\frac{1}{4}} = -\frac{1}{2} \)
• \( r_4 = \frac{-\frac{1}{16}}{\frac{1}{8}} = -\frac{1}{2} \)

Every ratio in this sequence equals \(-\frac{1}{2} \). This consistency confirms that \(-\frac{1}{2} \) is the common ratio. This means each term is simply the previous term multiplied by \(-\frac{1}{2} \).

Exploring Alternating Signs in Sequences

Alternating signs in a sequence mean that terms switch from positive to negative and vice versa. This can often be seen in sequences where the common ratio is negative. If the first term is negative, the next term will be positive when multiplied by a negative common ratio, and this pattern will continue.
For our sequence, the common ratio is \(-\frac{1}{2} \), which causes the signs to alternate:

• The first term is \(-1\)
• The second term becomes positive: \(\frac{1}{2} \)
• The third term switches back to negative: \(-\frac{1}{4} \)
• And the pattern continues, alternating with each term.

This alternating signs pattern is a key indicator of the negative common ratio in a geometric sequence.

Fraction Series in Geometric Sequences

A fraction series involves terms that are fractional numbers, which can add an extra layer of complexity. In our exercise, terms are fractions with alternating signs, creating a visually interesting sequence.
When dealing with fraction series in geometric sequences, it's important to remember:

• Consider the denominators of fractions; they can often reveal a power pattern.
• Calculate the ratios carefully to ensure you handle negative signs.

In this specific sequence, each term's denominator follows a power of 2:

• The first term, \(-1\), aligns with \(-\frac{2^0}{1} \)
• The second term, \( \frac{1}{2} \), is \(-\frac{2^1}{2} \)
• The third term, \(-\frac{1}{4} \), becomes \(-\frac{2^2}{4} \)
• This pattern continues through the sequence.

Understanding the underlying structure of the fractions helps in predicting subsequent terms and exploring the sequence further.