Problem 21 - College Algebra

Question

Use the formula for the sum of the first 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 Variables in Formula

In a geometric sequence, the formula for the sum of the first \( n \) terms is given by \( S_n = a_1 \frac{1-r^n}{1-r} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a_1 = 64 \) and \( r = 0.2 \), and \( n = 11 \).

Step 2: Substitute into Formula

Insert the values into the sum formula: \( S_{11} = 64 \frac{1-0.2^{11}}{1-0.2} \).

Step 3: Calculate the Power Term

Calculate \( 0.2^{11} \): \( 0.2^{11} = 2.048 \times 10^{-8} \).

Step 4: Evaluate the Formula

Substitute \( 0.2^{11} \) into the equation: \( S_{11} = 64 \frac{1-2.048 \times 10^{-8}}{0.8} \).

Step 5: Simplify the Expression

Simplify the numerator: \( 1 - 2.048 \times 10^{-8} \approx 1 \), thus \( S_{11} = 64 \times \frac{1}{0.8} \approx 80 \).

Key Concepts

Sum of Geometric Series

The sum of a geometric series is an important concept in mathematics, especially when dealing with sequences that involve constant ratio changes. This is essentially the accumulation of numbers in a sequence where each term is a fixed multiple of the previous one.
To find the sum of the first few terms in a geometric sequence, we use the sum formula:

• \( S_n = a_1 \frac{1-r^n}{1-r} \)

Here, \( S_n \) represents the sum of the first \( n \) terms, \( a_1 \) is the initial term of the series, \( r \) is the common ratio between consecutive terms, and \( n \) denotes the number of terms you intend to sum.
For our specific problem, the series sum \( S_{11} \) was determined by placing the given values into the formula. Through a series of calculations, including determining \( 0.2^{11} \), we found that the sum of the series, \( \sum_{a=1}^{11} 64 \cdot 0.2^{a-1} \), was approximately 80.

Common Ratio

The common ratio in a geometric sequence is a vital element to understand as it dictates how each term in the series is related to the previous term. It is the factor by which each term is multiplied to get the next one in the sequence.
Mathematically, it is represented as \( r \) and is found by dividing any term in the sequence by the preceding term. In our example:

• The common ratio \( r = 0.2 \).

This means that each term is 0.2 times the term before it. In practice, calculating the common ratio allows us to use the geometric series formula effectively to determine sums and specific terms within the sequence.
Recognizing the common ratio is the initial step in solving any problem related to geometric sequences, as it affects every subsequent calculation.

Geometric Series Formula

The geometric series formula is a powerful tool used to find the sum of terms in a geometric sequence. Its utility comes from its ability to handle sequences with a constant ratio and provides a quick method to calculate totals that might otherwise be cumbersome to add manually.
This formula is depicted as:

• \( S_n = a_1 \frac{1-r^n}{1-r} \)

It simplifies what could be a massive calculation into a single expression. By plugging in values for \( a_1 \) (first term), \( r \) (common ratio), and \( n \) (number of terms), you can effortlessly compute the sum. In our earlier exercise, we observed how each variable was substituted into the formula, which helped us arrive at the final calculated result.
Understanding how to apply this formula can make solving geometric problems more straightforward, ensuring clarity in results, especially when dealing with large sequences.