Problem 35 - A Graphical Approach to College Algebra

Question

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$\sum_{i=1}^{6} 5(2)^{i-1}$$

Step-by-Step Solution

Answer

The sum of the first six terms is 315.

Step 1: Identify the Components of the Formula

The sum of the first n terms of a geometric sequence is given by the formula \( S_n = a \frac{(r^n - 1)}{r - 1} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. For this problem, \( a = 5 \), \( r = 2 \), and \( n = 6 \).

Step 2: Calculate the Common Ratio Power

Compute \( r^n \) as part of the formula. Here, \( r = 2 \) and \( n = 6 \), so \( r^n = 2^6 = 64 \).

Step 3: Apply the Formula

Substitute the values into the sum formula: \( S_6 = 5 \frac{(64 - 1)}{2 - 1} \).

Step 4: Simplify the Expression

Simplify the expression inside the formula: \( S_6 = 5 \frac{(63)}{1} = 5 \times 63 \).

Step 5: Compute the Final Sum

Multiply to find the sum: \( S_6 = 5 \times 63 = 315 \).

Key Concepts

Understanding the Sum of a Geometric Sequence

In mathematics, a geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. To find the sum of the first several terms in a geometric sequence, we use an important formula. This formula is useful because it allows us to find the total value of a series of terms without manually adding each term individually, which can be convenient for sequences with many terms.

To find the sum of the first n terms of a geometric sequence, the formula is:\[S_n = a \frac{(r^n - 1)}{r - 1}\]where:

• \(S_n\) is the sum of the first n terms,
• \(a\) is the first term of the sequence,
• \(r\) is the common ratio, and
• \(n\) is the number of terms.

By substituting the respective values into this formula, you can easily calculate the sum of the sequence. For example, if \(a = 5\), \(r = 2\), and \(n = 6\), you follow the steps to find the sum of the first six terms.

Exploring the Geometric Series Formula

The geometric series formula is an elegant and efficient tool for managing sequences. Its significance is paramount in both academic and practical applications. Let’s dissect its components using a clear example, which will help demystify how it operates.

The formula \(S_n = a \frac{(r^n - 1)}{r - 1}\) takes into account the basic properties of a geometric sequence:

• The first term \(a\) sets the starting point.
• The common ratio \(r\) determines how each term progresses from the last.
• The exponent \(n\) raises \(r\) to the power, indicating exponential growth with each sequence term.

For example, if you are given a sequence where \(a = 5\), \(r = 2\), and you need the sum for the first 6 terms, this formula simplifies the process greatly. Compute \(r^n\) (i.e., \(2^6 = 64\)), substitute back to get \(S_n = 5 \frac{(64 - 1)}{2 - 1}\), and you swiftly find that the sum is 315. Understanding this step-by-step makes applying the formula in various scenarios much more intuitive.

Grasping the Common Ratio in Sequences

The common ratio in a geometric sequence is a fundamental concept and an important characteristic that defines the progression of terms. It determines how much each term of the sequence increases or decreases compared to the one before it.

Here’s how you can understand and use the common ratio:

• The common ratio \(r\) is obtained by dividing any term in the sequence by the previous term.
• In the sequence where each term is made by multiplying the previous term, the constant factor across terms is the common ratio.
• If \(r > 1\), each term increases compared to the last. Conversely, if \(0 < r < 1\), each term decreases.
• An \(r\) that equals zero would make the sequence converge to zero rapidly, depending on the sequence design.

In the given example with \(r = 2\), the terms are doubling each time. Hence, recognizing the power of the common ratio in predicting and calculating how sequences behave is vital for manipulating and understanding them effectively.