Problem 66 - College Algebra with Modeling and Visualization

Question

Write out the terms of the series and then evaluate it. $$\sum_{k=1}^{4} 5(2)^{k-1}$$

Step-by-Step Solution

Answer

The series evaluates to 75.

Step 1: Understanding the Series

The given series is \ \( \sum_{k=1}^{4} 5(2)^{k-1} \). This means that we need to find the sum of terms where each term has the form \ \( 5(2)^{k-1} \). The variable \( k \) will take integer values starting from 1 and ending at 4.

Step 2: Calculating Individual Terms

You will substitute the values of \( k \) from 1 to 4 into the expression \( 5(2)^{k-1} \) to find each term.- For \( k=1 \): \( 5(2)^{1-1} = 5(2)^{0} = 5 \times 1 = 5 \)- For \( k=2 \): \( 5(2)^{2-1} = 5(2)^{1} = 5 \times 2 = 10 \)- For \( k=3 \): \( 5(2)^{3-1} = 5(2)^{2} = 5 \times 4 = 20 \)- For \( k=4 \): \( 5(2)^{4-1} = 5(2)^{3} = 5 \times 8 = 40 \)

Step 3: Summing the Series

Now, add up all the terms you calculated:\[ 5 + 10 + 20 + 40 = 75 \].

Step 4: Final Evaluation

The value of the series \( \sum_{k=1}^{4} 5(2)^{k-1} \) is 75.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers where each term increases by a constant amount. This constant "step" between numbers is called the common difference. Imagine starting with a number and adding the same amount repeatedly; that's essentially what an arithmetic series is. For example, the series 2, 5, 8, 11, has a common difference of 3.

To find the sum of an arithmetic series, you can use the formula: \[ S_n = \frac{n}{2}(a + l) \] where:

• \( S_n \) is the sum of the series.
• \( n \) is the number of terms.
• \( a \) is the first term.
• \( l \) is the last term.

By using this formula, you add the first and last terms of the series and multiply that sum by half the number of terms. This simplifies the process of adding up each number separately, providing a quick way to find the total of a long sequence.

Geometric Series

A geometric series is a sequence where each term is found by multiplying the previous term by a fixed number, known as the common ratio. This type of series grows much faster or slower than an arithmetic series due to this multiplicative factor. Consider the series 3, 9, 27, 81, where each term is multiplied by 3.

To find the sum of a geometric series, especially when it's finite, use the formula: \[ S_n = a \frac{1-r^n}{1-r} \] where:

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

It's crucial to note that this formula applies when the absolute value of the common ratio \( |r| \) is not equal to 1. The series grows or shrinks based on whether \( r \) is greater than, less than, or equal to 1.

Summation Notation

Summation notation is a compact way of writing the sum of a series of terms. It's very useful for expressing complex series in a simplified manner. The notation uses the Greek letter sigma \( \Sigma \). You'll see something like this: \[ \sum_{k=1}^{n} a_k \] Here, \( k \) is the index of summation which runs from the lower limit (often 1) to the upper limit \( n \). Each term in the series is represented by \( a_k \), a general expression involving \( k \).

Using summation notation helps in understanding and computing the sums efficiently in problems involving a large number of terms. It provides a clear structure, indicating exactly how many terms to consider and what operation to perform on them. This neat way of notation is particularly advantageous in more advanced mathematics and helps to approach problems systematically.