Problem 57 - Beginning and Intermediate Algebra

Question

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}$$

Step-by-Step Solution

Answer

The sum of the terms of the given geometric sequence is \(S_5 = \frac{363}{243}\).

Step 1: Identify the first term, common ratio, and number of terms

The sequence given is $$1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}$$ The first term (a) is 1. To find the common ratio (r), we divide the second term by the first term, i.e., \(\frac{1}{3} \div 1 = \frac{1}{3}\). The number of terms (n) is 5, as there are 5 terms in the sequence.

Step 2: Apply the formula for the sum of a geometric sequence

Now that we have the values of a, r, and n, we can plug them into the formula for the sum of a geometric sequence: $$S_n = \frac{a(r^n -1)}{r-1}$$ We have a = 1, r = \(\frac{1}{3}\), and n = 5, so the formula becomes: $$S_5 = \frac{1 \left[\left(\frac{1}{3}\right)^5 -1 \right]}{\frac{1}{3}-1}$$

Step 3: Simplify the formula and compute the sum

Now we will simplify the formula and calculate the sum: $$S_5 = \frac{1 \left[\frac{1}{243} - 1\right]}{-\frac{2}{3}}$$ Then, we find the common denominator to simplify the fraction inside the brackets: $$S_5 = \frac{-\frac{242}{243}}{-\frac{2}{3}}$$ To divide these fractions, we flip the second fraction (the divisor) and multiply the fractions: $$S_5 = -\frac{242}{243} \times \frac{3}{-2} = \frac{242}{243} \times \frac{3}{2}$$ Finally, multiply the numerators and the denominators to get the sum: $$S_5 = \frac{242 \times 3}{243 \times 2} = \frac{726}{486}$$ This fraction can be simplified to: $$S_5 = \frac{363}{243}$$ Thus, the sum of the terms of the given geometric sequence is \(\frac{363}{243}\).

Key Concepts

Sum of Geometric Series

A geometric series is simply the sum of the terms in a geometric sequence. If you add up all the terms from a specific geometric sequence, you get what is known as the sum of that series. Understanding how to calculate this is crucial, especially when you're dealing with a large number of terms. The sum of a geometric series can be calculated using a specific formula, which helps simplify what could otherwise be a quite lengthy calculation. Simply put: the key idea is to use the formula, which makes the process of summing very efficient and ensures accuracy.

Common Ratio

In a geometric sequence, the 'common ratio' is what defines the sequence itself. This is the factor that you multiply by each term to get to the next term. In our example sequence, \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}\), the common ratio is identified by dividing any term by its preceding term. For example, \(\frac{1}{3} \div 1 = \frac{1}{3}\). This ratio remains consistent across all terms in the sequence and is crucial for identifying the characteristic of the sequence. It helps determine the rule governing the sequence, which is vital to solving problems related to geometric sequences.

Sum Formula

The formula used to find the sum of a geometric series is:

• \(S_n = \frac{a(r^n -1)}{r-1}\)

Where:

• \(a\) is the first term in the sequence,
• \(r\) is the common ratio,
• \(n\) is the number of terms you'd like to sum.

Understanding each component is essential. By plugging the known values into this formula, you can effortlessly find the sum of any finite geometric series. Ensure you simplify step-by-step, especially when dealing with fractions, as this aids in accuracy. It’s a powerful tool that quickly transforms complex problems into straightforward solutions.

Sequence Terms

The terms of a sequence are the individual numbers or elements that make up the sequence. In a geometric sequence, these terms are generated by repeatedly multiplying the first term by the common ratio. For example, in the sequence \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}\), each term is acquired by multiplying the previous term by \(\frac{1}{3}\). Each element in the sequence plays a vital role, particularly when calculating the sum of the series. Recognizing the order and value of these terms is important in understanding the sequence altogether, as it helps you apply the appropriate formulas and ensures you're following the correct steps in problem-solving.