Problem 71 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{i=1}^{7}64\left(-\dfrac{1}{2}\right)^{i - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is approximately 52.73.

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

For the given finite geometric series, the first term (a) is 64. The common ratio (r) is \( -\dfrac{1}{2} \). The number of terms (n) is 7.

Step 2: Substitute the values into the formula

Substitute a = 64, r = \( -\dfrac{1}{2} \), and n = 7 into the formula for the sum of a finite geometric series, \( S = \frac{a \cdot (r^n - 1)}{r - 1} \). So \( S = \frac{64 \cdot \left((-1/2)^7 - 1\right)}{-1/2 - 1} \).

Step 3: Simplify the equation

First calculate the value of \( r^n \), that is \( (-1/2)^7 = -0.0078125 \). Then substitute this into the formula and compute to find \( S = \frac{64 \cdot (-0.0078125 - 1)}{-1/2 - 1} = 52.7265625 \).

Key Concepts

Finite Geometric Series

When working with a finite geometric series, you're dealing with a specific type of sequence. This sequence has a finite number of terms, which makes arithmetic calculations more manageable. The terms in a geometric series are produced by multiplying the previous term by a constant, known as the common ratio. A finite geometric series, therefore, is the sum of these terms. It is crucial to identify how many terms you have, the first term, and the common ratio to find the sum. In the example provided, there are 7 terms in the series, with the starting term being 64 and the common ratio being \(-\frac{1}{2}\).

• Identify the first term.
• Determine the number of terms.
• Recognize the pattern dictated by the common ratio.

The finite nature of this series allows it to be summed using the sum formula, which we'll describe next.

Sum Formula

To find the sum of a finite geometric series, we use a specific formula. This formula makes it easy to calculate the total. The sum formula for a finite geometric series is:\[ S = \frac{a \cdot (r^n - 1)}{r - 1} \]where:

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

In our exercise, using 64 as the first term, \(-\frac{1}{2}\) as the common ratio, and 7 as the number of terms, we substitute these into the formula:\[ S = \frac{64 \cdot \left((-1/2)^7 - 1\right)}{-1/2 - 1} \]By simplifying, you arrive at the total sum of the series, which in this example is approximately 52.73.

Common Ratio

The common ratio in a geometric series is the factor by which you multiply each term to get the next one. Understanding this concept is crucial for analyzing how the series expands or contracts. The common ratio can be any real number, including negative values. In our example, the common ratio is \(-\frac{1}{2}\). This tells us that each term is half of the previous term and that the sign alternates between positive and negative.

• If the common ratio is larger than 1, the series grows quickly.
• If the common ratio is between 0 and 1, the series decreases or grows slowly.
• If it is negative, expect alternating signs in the series.

Knowing the common ratio helps predict the behavior of the series and is critical when applying the sum formula. This understanding allows you to correctly compute the sum of finite geometric series as demonstrated in the exercise.