Problem 6 - Algebra 2

Question

Evaluate the finite series for the specified number of terms. $$ -\frac{1}{6}+1-6+36-\ldots ; n=5 $$

Step-by-Step Solution

Answer

The value of the sum of the series up to the fifth term is \(-1111\).

Step 1: Identify the common ratio

The common ratio \(r\) for a geometric series is found by dividing any term by its preceding term. Given that our sequence begins as \(-\frac{1}{6},1,-6,36...\), we can safely say the common ratio \(r\) is \(-6\).

Step 2: Identify the first term

The first term \(a\) of our series is clearly stated as \(-\frac{1}{6}\).

Step 3: Apply the formula for sum of finite geometric series

The formula for the sum \(S\) of a finite geometric series is \(S = a \times \frac{1 - r^n}{1 - r}\) where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Substitute into the formula \(a = -\frac{1}{6}\), \(r = -6\) and \(n = 5\).

Step 4: Simplify the expression

After substituting you will get \(S = -\frac{1}{6} \times \frac{1 - (-6)^5}{1 - -6}\). This simplifies to \(S = -\frac{1}{6} \times \frac{1 - 7776}{7}\). Arithmetically processing this expression leads to the final answer.

Key Concepts

Finite Series

A finite series is a sum of a sequence that has a specific number of terms. Unlike an infinite series, which continues indefinitely without terminating, a finite series has an end. In this context, our series is geometric and made up of 5 specific terms as mentioned in the exercise.
\(-\frac{1}{6} + 1 - 6 + 36 - \ldots\)

• A finite series with a set number of terms is often easier to evaluate as compared to infinite series, since you can count and apply calculations on a manageable number of elements.
• When dealing with a finite series, the final answer is straightforward to identify after all elements have been summed up or calculated using a formula.

For students, understanding finite series forms the basis for grasping more complex sequences and series problems. You can think of finite series as a closed, complete series where each term is crucial for the final sum.

Common Ratio

The common ratio in a geometric series is a consistent factor that you multiply with one term to get the next term in the series. Finding the common ratio is critical for understanding the progression of the series.
For the series given: \(-\frac{1}{6}, 1, -6, 36, \ldots\), the common ratio \(r\) is \(-6\). To find it:

• Pick any term and divide it by the term that comes right before it.
• For instance, dividing 1 by \(-\frac{1}{6}\) gives \(-6\).

Working with the right common ratio is imperative as it plays a pivotal role in the formula to find the sum of the series. In geometric sequences, this factor aids in calculating subsequent terms when the starting point or first term is known.

Sum of Series

The sum of a geometric series, especially a finite one, can be calculated using a specific formula:
\[ S = a \times \frac{1 - r^n}{1 - r} \]
Where:

• \(S\) is the sum of the series.
• \(a\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the number of terms.

Applying this formula allows you to find the total of the terms in the series without manually adding each term, which is particularly handy with complex series.
Let's do a quick application with our example:

• Given \(a = -\frac{1}{6}\), \(r = -6\), and \(n = 5\).
• Plug these values into the formula to calculate the sum \(S\).
• The expression simplifies into \(-\frac{1}{6} \times \frac{1 - (-6)^5}{7}\).

Understanding how to utilize this formula effectively is valuable for solving series problems in mathematics, and it can also help in various fields where pattern predictions and calculations of consecutive entities are required.