Problem 55 - College Algebra

Question

Express each sum using summation notation. Use a lower limit of summation of your choice and \(k\) for the index of summation. $$5+7+9+11+\dots+31$$

Step-by-Step Solution

Answer

\[\sum_{k=1}^{14} (5 + 2(k - 1))\]

Step 1: Identify the first term, common difference, lower and upper limit

The first term (A) is 5, the common difference (d) is 2 since each term increases by 2, the lower limit (a) will be chosen as 1 so that i = 1 gives us the first term 5, and the upper limit (b) will be calculated by substituting A, d, a and the last term into the series formula \[A + d(b - a) = last-term\] to solve for b.

Step 2: Calculate the upper limit

Substitute A = 5, d = 2, a = 1, last-term = 31 into the formula:\[5 + 2(b - 1) = 31\]which gives:\[2b = 31 - 5 + 2 = 28\]Solving for b gives b = 14

Step 3: Express the series using the summation notation

Now we can write the arithmetic series in a summation notation:\[\sum_{k=1}^{14} (5 + 2(k - 1))\]

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference. The series begins at a starting number and continues by adding this constant to generate the next terms. In our example, the numbers 5, 7, 9, 11, ..., 31 form an arithmetic series because each number increases by the same amount.

Key characteristics of an arithmetic series include:

• The first term (denoted as \( A \)) is usually the starting point of the series. Here, the first term is 5.
• The common difference (denoted as \( d \)) is the consistent increment between consecutive terms. In this example, \( d = 2 \).
• The sequence continues until a specified upper limit, here reaching up to 31.

Stay tuned to learn more about the common difference, one of the core aspects that forms the foundation of this arithmetic series.

Common Difference

The common difference in an arithmetic series is a crucial component. It is the amount added to each term to arrive at the next term. In our referenced series, the difference between each consecutive pair of numbers is 2. This means that every number is 2 greater than the one before it.

Knowing the common difference allows us to calculate any term in the series without listing all of its elements. For example, starting from the first term 5, you add the common difference multiple times to generate subsequent terms:

• 1st term: 5
• 2nd term: 5 + 2 = 7
• 3rd term: 7 + 2 = 9

This consistent difference is what makes it possible to write the series in a formulaic way using summation notation, which we'll address shortly in the series formula section.

Series Formula

The series formula is used to determine specific properties of an arithmetic series, such as the upper limit in a sequence, based on its terms. For our example, the series formula is expressed as \[ A + d(b - a) = ext{last-term} \] where:

• \( A \) is the first term (5 in this case)
• \( d \) is the common difference (2)
• \( a \) is the index of the first term, often starting at 1
• \( b \) is the index of the last term, which needs to be determined

To find the upper limit, we substitute the values: \[ 5 + 2(b - 1) = 31 \]Simplifying gives us:\[ 2b = 28 \]\[ b = 14 \]This value represents the number of terms needed to reach the final term in the series from the first, enabling us to form the index of summation effectively.

Index of Summation

The index of summation is a critical concept in expressing an arithmetic series in summation notation. It indicates the range over which you sum the sequence’s terms. In our series, the index of summation is represented by the variable \( k \).

In the context of our example, the summation notation is expressed as: \[ \sum_{k=1}^{14} (5 + 2(k - 1)) \] This notation indicates that \( k \) starts at 1 and ends at 14. For each integer value of \( k \), the expression \( 5 + 2(k-1) \) yields each term of the arithmetic series.

The summation notation helps in compactly writing down the series while specifying exactly how many terms are included, and it gives you a systematic way to handle series in mathematical problems. Understanding this notation is essential for anyone working with series and wanting to describe them efficiently.