Problem 55 - Beginning and Intermediate Algebra

Question

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$-\frac{1}{4},-\frac{1}{2},-1,-2,-4,-8$$

Step-by-Step Solution

Answer

The sum of the terms in the given geometric sequence \(-\frac{1}{4},-\frac{1}{2},-1,-2,-4,-8\) is \(S_6 = \frac{63}{4}\).

Step 1: Determine the common ratio between terms

In a geometric sequence, the common ratio (r) is the factor that we multiply each term by to get the next term. We can find the common ratio by dividing any term by the term before it. For example: r = \(\frac{-1/2}{-1/4} = 2\) Therefore, the common ratio (r) is 2.

Step 2: Find the total number of terms (n) in the sequence

Count the number of terms in the given sequence. There are 6 terms in this sequence: \(-\frac{1}{4},-\frac{1}{2},-1,-2,-4,-8\) So, n = 6.

Step 3: Use the formula for \(S_n\) to find the sum

Now we have all the values we need to use the formula for the sum of the first n terms of a geometric sequence: \[S_n = \frac{a_1(1 - r^n)}{1 - r}\] Plug in the values: \(S_6 = \frac{-\frac{1}{4}(1 - 2^6)}{1 - 2}\)

Step 4: Calculate the sum

Now, we just simplify the expression and solve for the sum: \(S_6 = \frac{-\frac{1}{4}(1 - 64)}{-1}\) \(S_6 = -\frac{1}{4}(-63)\) \(S_6 = \frac{63}{4}\) Thus, the sum of the terms in the given geometric sequence is \(\frac{63}{4}\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the key number that determines the relationship between each consecutive term. To find this ratio, you simply take any term in the sequence and divide it by the preceding term. For example, if you have a sequence

• Term 1: \(-\frac{1}{4}\),
• Term 2: \(-\frac{1}{2}\),
you would find the common ratio by calculating:\[ r = \frac{-\frac{1}{2}}{-\frac{1}{4}} = 2 \]This tells us that each term in the sequence is twice the preceding term. Knowing the common ratio is crucial to understanding how the sequence progresses. It allows you to predict subsequent terms and plays a central role in calculating the sum of the series. Whenever you are working on a geometric sequence, identifying the common ratio is your first step.

Sum of Geometric Series

The sum of a geometric series refers to the total of all terms in the sequence up to a certain point. To calculate this, you use the formula:\[ S_n = \frac{a_1(1 - r^n)}{1 - r} \]where:

• \(S_n\) is the sum of the first \(n\) terms,
• \(a_1\) is the first term of the sequence,
• \(r\) is the common ratio,
• \(n\) is the number of terms.
For the sequence \(-\frac{1}{4}, -\frac{1}{2}, -1, -2, -4, -8\), we calculated:\[ S_6 = \frac{-\frac{1}{4}(1 - 2^6)}{1 - 2} \]This expression simplifies to:\[ S_6 = \frac{63}{4} \]The formula is powerful because it allows for quick computation of the sum without individually adding each term. Understanding how to apply this formula ensures that you can efficiently solve problems involving geometric series. It’s particularly useful in financial models, signal processing, and other mathematical applications.

Sequence Terms

Sequence terms are the individual components of a sequence. In a geometric sequence, each term follows from the previous one by multiplying by a fixed, non-zero number called the common ratio. For example, in the sequence

• \(-\frac{1}{4}\),
• \(-\frac{1}{2}\),
• \(-1\),
• \(-2\), etc.,
each term can be obtained by multiplying the previous term by the common ratio of 2. Geometric sequences are defined by this consistent multiplier, and the understanding of each term's relationship within the sequence is fundamental. It's important because it lays the groundwork for more complex operations, like finding the sum or predicting future terms in the sequence. By examining each term and understanding its place, you also develop a deeper insight into the structure of the sequence as a whole.