Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, 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 is 80.

Step 1: Identify the terms in the geometric sequence

In a geometric sequence, each term is obtained by multiplying the previous term by a constant known as the common ratio. Here, the sequence is represented by \( a_n = 64 \cdot 0.2^{n-1} \). This means the first term \( a_1 = 64 \) and the common ratio \( r = 0.2 \).

Step 2: Recognize the formula for the sum of a geometric series

The sum of the first \( n \) terms of a geometric sequence is given by the formula:\[S_n = a_1 \frac{1-r^n}{1-r}\]where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 3: Substitute values into the sum formula

We need to find the sum of the first 11 terms, so \( n = 11 \). Plug in the values: \( a_1 = 64 \), \( r = 0.2 \), and \( n = 11 \): \[S_{11} = 64 \frac{1-(0.2)^{11}}{1-0.2}\]

Step 4: Calculate the value of \( r^n \)

Calculate \( 0.2^{11} \):\[0.2^{11} = 2.048 \times 10^{-8}\]This value is effectively zero when used in calculations due to its small size.

Step 5: Compute the sum \( S_n \)

Substituting \( 0.2^{11} \approx 0 \) and calculating the expression:\[S_{11} = 64 \frac{1-0.00000002048}{0.8} \approx 64 \frac{1}{0.8} = 64 \times 1.25 = 80\]

Step 6: State the sum of the series

The sum of the first 11 terms of the given geometric sequence is 80.

Key Concepts

Sum of a Geometric Series

The sum of a geometric series refers to the total of the first few terms of a sequence where each subsequent term is a consistent multiple of the previous one. To determine this sum, we can use a special 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 first term in the sequence, and \( r \) is the common ratio.

When calculating, a crucial aspect is recognizing whether \( |r| < 1 \) as it significantly simplifies our calculations because terms diminish rapidly. In this exercise, we see that the sum of the first 11 terms evaluates to 80. This is largely because the factor \( r^{11} \), given \( r \) is 0.2, becomes negligible and approaches zero, making it simpler to compute the final sum.

Common Ratio

A central component of a geometric sequence is the common ratio, often denoted as \( r \). This ratio determines how the sequence progresses by controlling the relationship between consecutive terms. To clarify:

• The common ratio is found by dividing any term in the sequence by the previous one.
• In our sequence, since \( a_n = 64 \times 0.2^{n-1} \), the common ratio \( r \) is 0.2, a positive fraction.

Understanding the common ratio helps in predicting the pattern of growth or decay in the sequence:

• If \( r > 1 \), the sequence grows as each term becomes larger than the last.
• If \( r = 1 \), every term is the same, resulting in a constant series.
• If \( 0 < r < 1 \), which is the case here, the sequence shrinks and each term is smaller than the last, approaching zero.

This insight into the common ratio is vital to manipulating and understanding geometric sequences.

First Term in a Geometric Sequence

The first term of a geometric sequence, often denoted as \( a_1 \), serves as the foundational starting point from which all subsequent terms are calculated. It sets the stage for the initial value before any multiplication by the common ratio takes place.

• In our exercise, the first term \( a_1 \) is 64.
• This initial value is crucial as it heavily influences the overall sum of the series.

Every series will look different based on its first term and common ratio. A high first term means that the subsequent values of the sequence will be significant unless the common ratio is very small. Understanding the impact of the first term allows us to better appreciate how sequences are structured and how their sums are formed.