Question

Is the series geometric? If so, give the number of terms and the ratio between successive terms. If not, explain why not. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\cdots+\frac{1}{256} $$

Short answer

If so, what are the common ratio and total number of terms in the series? Answer: Yes, the series is geometric with a common ratio of $$- \frac{1}{2}$$. There are a total of 9 terms in the series.

Step-by-step solution

Identify the Pattern of the Series

Firstly, we observe the sequence of terms in the series: $$1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, \dots, \frac{1}{256}.$$ We notice that each term seems to alternate in sign, switching between positive and negative values.

Check if the Common Ratio is Consistent

To check if the series is geometric, we divide each term by the preceding term. $$\frac{-\frac{1}{2}}{1} = -\frac{1}{2}, \frac{\frac{1}{4}}{-\frac{1}{2}} = -\frac{1}{2}, \frac{-\frac{1}{8}}{\frac{1}{4}} = -\frac{1}{2}, \dots.$$ We can see that the common ratio between successive terms is consistently $$- \frac{1}{2}$$.

Count the Number of Terms

Now we need to find the number of terms of the series. Since the series ends at $$\frac{1}{256}$$, let's find how many times we multiply the first term, 1, by the common ratio $$- \frac{1}{2}$$ to get the last term. $$1 \cdot \left(-\frac{1}{2}\right)^{n - 1} = \frac{1}{256}$$ Using the logarithm, we find: $$n - 1 = \log_{-\frac{1}{2}}{\frac{1}{256}} \Rightarrow n- 1 = 8 \Rightarrow n = 9.$$ So there are 9 terms in the series.

Conclusion

The given series $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\cdots+\frac{1}{256}$$ is a geometric series with a common ratio of $$- \frac{1}{2}$$ and a total of 9 terms.

Key concepts

Common Ratio

When dealing with a geometric series, one key aspect to determine is the "common ratio." This is the factor by which each term in the series is multiplied to get the next term.
Simply put, if you have a term in the series, you multiply it by the common ratio to find the next one.

In our exercise example, after analyzing the sequence \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, \ldots, \frac{1}{256}\), we observed how the terms alternate between positive and negative.
Each term is divided by the one before it to see if there is a consistent ratio.
The calculations reveal that each division results in \(-\frac{1}{2}\). Thus, \(-\frac{1}{2}\) is the common ratio.

This consistent multiplying factor confirms that it is indeed a geometric series. Always remember: recognizing a constant ratio is crucial when identifying geometric sequences.

Alternating Series

An alternating series is identified by a distinctive pattern where the series' terms switch between positive and negative.
This alternating nature creates a zigzag of terms in the overall sequence.

In the given series \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}\), the signs change between every two consecutive terms.
This is a clear hallmark of an alternating series.
The alternating sign often arises as part of the common ratio in geometric series with negative values.

Here, our common ratio \(-\frac{1}{2}\) results in the sequence alternation.
The negative sign is engaging when you consider how it flips each subsequent term's sign.
Suddenly, the algebra isn't just about numbers—it's about producing series with visually engaging patterns.

Number of Terms

The "number of terms" in a series defines how many elements are present in that sequence.
It outlines the series' scope from start to finish.

In geometric series, understanding how many times the common ratio is applied provides critical insight.
Using the given series, the task becomes finding how often you multiply the first term, which is \(1\), by \(-\frac{1}{2}\) to arrive at \(\frac{1}{256}\).
Mathematically, this is seen as: \[1 \cdot \left(-\frac{1}{2}\right)^{n-1} = \frac{1}{256}\] By solving \(n-1 = 8\), the calculation shows there are 9 terms.

Each term requires consistent application of the common ratio.
Therefore, pinning down the total number of terms verifies the series' boundaries and how it expands.
It ensures that you've correctly captured the full series from start to finish.