Problem 6 - College Algebra

Question

Express each description of a sum using summation notation. The sum of terms \(m^{2}+3 m\) from \(m=1\) to \(m=5\)

Step-by-Step Solution

Answer

\( \sum_{m=1}^{5} (m^2 + 3m) \)

Step 1: Identify the Summation Formula

First, recognize the expression that repeats in the sum. The expression given is \( m^2 + 3m \). This is the expression that we will use to create our summation notation.

Step 2: Determine the Range of Summation

Next, we need to determine the values that our index variable \( m \) will take. According to the problem, \( m \) starts at 1 and ends at 5. So, the range of \( m \) is from 1 to 5.

Step 3: Write the Summation Using Notation

We can now write the sum using summation notation. Start with the summation symbol \( \Sigma \). Below \( \Sigma \), write the starting point of \( m \), which is \( m=1 \). Above \( \Sigma \), write the ending point of \( m \), which is 5. To the right of \( \Sigma \), write the expression \( m^2 + 3m \). This gives us:\[ \sum_{m=1}^{5} (m^2 + 3m) \]

Key Concepts

Summation Formula

In mathematics, a summation formula is used to represent the sum of a sequence of numbers compactly. The essence of a summation formula is the expression that defines each term of the sequence. In the given exercise, the summation formula is the expression \( m^2 + 3m \). This represents the individual terms that contribute to the total sum as the index variable \( m \) changes its values.

The beauty of using a summation formula is it simplifies the description of the summation process. Instead of listing out each individual term manually, such as \( 1^2 + 3 \times 1 \), \( 2^2 + 3 \times 2 \), and so on, the summation formula wraps this entire process into a concise mathematical statement. This aids in both clarity and computational efficiency, making it an essential tool in algebra and calculus. Remember, understanding the essence of the expression within the formula is key to utilizing summation in various math problems.

Index Variable

An index variable is a placeholder that tells us which terms we include in our summation. In the context of the given problem, \( m \) serves as the index variable. It takes on a series of values, as defined by its range, which in this case goes from 1 to 5.

The use of an index variable like \( m \) is integral because it establishes:

• The starting point of the sequence of terms. Here, \( m \) starts at 1.
• The ending point, or the point where we stop calculating additional terms, which is 5 in this example.
• The actual sequence defined in the summation formula by substituting different values of \( m \).

The index variable steps through its specified range; for each integer value, it replaces \( m \) in the expression, generating terms for the sum. This sequential substitution is what enables us to calculate meaningful totals from complex expressions.

Summation Symbol

The summation symbol, denoted by \( \Sigma \), is a Greek letter that stands for 'sum' in mathematical notation. It signifies the operation of adding up a series of terms. In the summation notation, the symbol \( \Sigma \) is the centerpiece, around which the details of the summation are arranged.

Here's how it works in practice:

• The initial value for the index variable is placed below the \( \Sigma \). For our exercise, this is \( m = 1 \).
• The final value is positioned above the \( \Sigma \), which in this example is 5. This states the extent of the sequence.
• To the right of \( \Sigma \), you provide the expression involving the index variable, as seen with \( m^2 + 3m \).

Putting together these components results in a concise way to represent what could be a lengthy list of additions. The summation symbol \( \Sigma \) thereby simplifies complex arithmetic operations, allowing mathematicians and students alike to communicate ideas more neatly and calculate efficiently.