Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, 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: Understand the Description

The problem describes a sum of terms given by the expression \(m^2 + 3m\), where \(m\) varies from \(1\) to \(5\). This means we need to consider each term of the form \(m^2 + 3m\) for \(m = 1, 2, 3, 4, 5\).

Step 2: Identify the Sequence Formula

The expression given for each term, \(m^2 + 3m\), serves as the sequence formula. This means for each value of \(m\) between \(1\) and \(5\), we calculate the term using this formula.

Step 3: Set the Range of Summation

The range of summation starts from \(m = 1\) and ends at \(m = 5\). This tells us which integers we will substitute into the expression \(m^2 + 3m\) to compute each term in the sequence.

Step 4: Write the Summation Notation

Summation notation is a concise way to express the sum of a sequence of terms. Using the sequence formula and the specified range, the sum can be written as: \[\sum_{m=1}^{5} (m^2 + 3m)\]This describes the complete summation of the terms from \(m = 1\) to \(m = 5\).

Key Concepts

sequence formula

A sequence formula is like a recipe that tells us how to find each term in a sequence. For the given exercise, the formula is provided as \(m^2 + 3m\). This formula is crucial because it defines what each term in our sum will look like, when we plug in different values of \(m\).

The sequence formula is applicable for each member in the range specified by the summation. Here, as \(m\) takes values from 1 to 5, you'll calculate those terms by substituting each integer one at a time into the formula:

• When \(m = 1\), the term is \(1^2 + 3 \cdot 1 = 4\).
• When \(m = 2\), the term is \(2^2 + 3 \cdot 2 = 10\).
• When \(m = 3\), the term is \(3^2 + 3 \cdot 3 = 18\).
• When \(m = 4\), the term is \(4^2 + 3 \cdot 4 = 28\).
• When \(m = 5\), the term is \(5^2 + 3 \cdot 5 = 40\).

This sequence of calculations is fundamental when expressing sums in summation notation, ensuring that you understand what each term contributes to the sum.

range of summation

The range of summation determines which specific integer values of \(m\) you will use in your sequence formula. It's like the boundary lines of a field that tell us where we start and stop calculating.

In this exercise, the range is from \(m = 1\) to \(m = 5\). It specifies that we include every integer starting at 1, going up to and including 5. The notation used to represent this range in summation notation is \(\sum_{m=1}^{5}\). This is a powerful and convenient way to convey not just the formula you're using, but also exactly which terms you need.

• Starting Point: The range starts at 1, meaning include the term when \(m = 1\).
• Ending Point: It goes up to 5, so ensure you consider the term when \(m = 5\) as your last term.

Having a clear range is essential; it directs the computation and helps you in organizing the terms you need to sum up.

steps to express sums in notation

Expressing sums in summation notation is a methodical process that makes large sums easier to write and understand. Here is how you can express the sum using the given sequence formula and range of summation.

**Step 1: Identify the Sequence Formula** Begin by recognizing the sequence formula, which describes each term of the sum. In this case, it's \(m^2 + 3m\). This formula tells you how to find individual terms based on the changing \(m\) values.

**Step 2: Determine the Range of Summation** This involves identifying the starting and ending points of the summation, noted by \(m = 1\) to \(m = 5\) in the given problem. These bounds instruct on the iterations from the sequence formula to include in the sum. **Step 3: Compose the Summation Notation** Finally, combine these insights into the summation notation: \(\sum_{m=1}^{5} (m^2 + 3m)\). This expression represents the entire sum from \(m=1\) to \(m=5\), where each term is computed using \(m^2 + 3m\).
Summation notation provides a compact and efficient way to express both small and large sums, greatly simplifying the notation and clarity of the sum you're working with.