Problem 21 - College Algebra

Question

Use the formula for the sum of the fi st \(n\) terms of each geometric sequence, and then state the indicated sum. $$ \sum_{a=1}^{11} 64 \cdot 0.2^{a-1} $$

Step-by-Step Solution

Answer

The sum of the series is approximately 80.

Step 1: Identify the Formula

To find the sum of the first \(n\) terms of a geometric sequence, we use the formula \(S_n = a_1 \frac{1 - r^n}{1 - r}\) where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 2: Identify Parameters

From the given series \( \sum_{a=1}^{11} 64 \cdot 0.2^{a-1} \), we identify \(a_1 = 64\), \(r = 0.2\), and the number of terms \(n = 11\).

Step 3: Apply the Formula

Substitute \(a_1 = 64\), \(r = 0.2\), and \(n = 11\) into the formula: \[ S_{11} = 64 \frac{1 - (0.2)^{11}}{1 - 0.2} \].

Step 4: Simplify the Denominator

Calculate the denominator: \(1 - 0.2 = 0.8\).

Step 5: Calculate the Numerator Expression

Compute \(1 - (0.2)^{11}\). First, find \((0.2)^{11}\): - \((0.2)^{11} \approx 2.048 \times 10^{-8}\).Then, \(1 - (0.2)^{11} \approx 0.99999997952\).

Step 6: Calculate the Sum

Now, substitute back into the formula:\[ S_{11} = 64 \cdot \frac{0.99999997952}{0.8} \].Simplify this to find:\[ S_{11} \approx 64 \cdot 1.2499999744 \approx 79.9999983616 \].

Key Concepts

Sum of Geometric Series

A geometric series is a sum of the terms of a geometric sequence. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. To calculate the sum of the first few terms in a geometric sequence, we use the sum formula, which is super helpful. It allows you to find out how much the series adds up to for any number of terms.The formula for the sum of the first \(n\) terms of a geometric sequence is:\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]Here's a quick breakdown of what this formula means:

• **\(a_1\)** is the first term in the sequence.
• **\(r\)** is the common ratio.
• **\(n\)** is the number of terms that you want to add up.

This formula gives you the total of the entire sequence quickly and efficiently, rather than adding each term individually. For example, using the formula for our given sequence, we calculate the sum \(S_{11} = 79.9999983616\), showcasing how precise calculations can be with it.

Common Ratio

In a geometric sequence, the common ratio \(r\) is the factor you multiply each term by to get the next term. Finding the common ratio is straightforward. If you know two consecutive terms, just divide the later term by the earlier one, and that's your common ratio! It's vital since it tells you how quickly or slowly the sequence grows.For instance, in the sequence given in the exercise \(64, 12.8, 2.56,\)..., we have:

• **First term:** 64
• **Second term:** 12.8

Divide the second term by the first:\[ r = \frac{12.8}{64} = 0.2 \]This ratio, 0.2, indicates that each term is \(0.2\) times the previous term. The common ratio can be less than one, like in this example, equal to one, or more than one, which respectively make the sequence shrink, stay constant, or grow with each term.

Number of Terms

The number of terms \(n\) in a geometric sequence determines how many terms you're dealing with when calculating the sum using the formula. Identifying this number is usually the first step in solving problems related to geometric series. The term "number of terms" simply refers to how many numbers there are in the sequence you're summing up.In our exercise, it is stated clearly that we must find the sum of the first 11 terms. Thus, \(n = 11\). This means we start from the first term and count all the way to the eleventh term, without skipping any. By using the formula "\(S_n = a_1 \frac{1 - r^n}{1 - r}\)," this count becomes crucial, especially when calculating powers and making sure you reach the correct end to your geometric sequence calculations.

First Term

The first term of a geometric sequence, denoted as \(a_1\), is the starting point of the sequence. It's the initial value that we build upon using the common ratio. Knowing the first term is crucial because it sets the base for all subsequent terms in the sequence.In our specific example, the first term \(a_1\) is 64. This means the sequence starts with 64, and each following term is calculated by multiplying 64 by increasingly higher powers of the common ratio. The first term is often directly given or is the initial figure given in a word problem. It plays a key role in the sum formula \(S_n = a_1 \frac{1 - r^n}{1 - r}\), showing its necessity both in sequence creation and in sum calculations.