Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, express each description of a sum using summation notation. The sum of \(6 k-5\) from \(k=-2\) to \(k=1\)

Step-by-Step Solution

Answer

\(\sum_{k=-2}^{1} (6k - 5)\)

Step 1: Understand the Problem

The problem requires expressing the sum of a sequence defined by the expression \(6k - 5\) for values of \(k\) ranging from \(-2\) to \(1\) using summation notation.

Step 2: Identify the General Formula and Limits

Recognize that the expression for the sequence is \(6k - 5\), and the sequence starts at \(k = -2\) and ends at \(k = 1\). The limits for the summation are from \(k = -2\) to \(k = 1\).

Step 3: Write the Summation Notation

Using the identified formula and limits, the sum can be written in summation notation as: \[\sum_{k=-2}^{1} (6k - 5)\] This notation succinctly represents the sum of the sequence calculated for each integer value of \(k\) from \(-2\) to \(1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They form the building blocks of more complex mathematical concepts and are essential for representing mathematical relationships and equations.
For example, consider the expression \(6k - 5\). In this expression, \(6k\) indicates that the variable \(k\) is multiplied by 6, and then 5 is subtracted from the result. Algebraic expressions can represent quantities and changes; they are versatile tools used to translate real-world scenarios into mathematical form.
When using these expressions, it’s crucial to understand each component:

• Constants: These are fixed values. In our example, -5 is a constant.
• Coefficients: Numbers that multiply a variable, like 6 in the term 6k.
• Variables: Symbols that stand for unknown values. Here, \(k\) is a variable.

Recognizing how these elements interact allows us to manipulate and solve expressions effectively.

Series Notation

Series notation, often represented with sigma (\(\Sigma\)) notation, is a concise way to denote the sum of a sequence of numbers. It is used to simplify the writing and understanding of long sequences of additions. In our specific problem, the series notation is represented as \( \sum_{k=-2}^{1} (6k - 5) \).
This notation tells us several things instantly:

• Summand: The expression \(6k - 5\) is what you sum over the specified range.
• Lower limit: The starting point \(k = -2\) denotes where the summation begins.
• Upper limit: The ending point \(k = 1\) indicates where the summation stops.

The series notation functions as a mathematical shorthand to convey the sum of values derived by evaluating the function inside the summand across all integer values between the lower and upper limits. It is particularly handy in complex calculations where clarity and brevity are needed.

Mathematical Sequences

A mathematical sequence is a ordered list of numbers following a specific rule or pattern. Each number in the sequence is called a term. In the exercise, the sequence is generated using the expression \(6k - 5\) for values of \(k\) from -2 to 1.
Let's break down how sequences work:

• Rule of Formation: The sequence is defined by a formula, such as \(6k - 5\). This formula gives us a way to generate terms by substituting different values for \(k\).
• Term Position: Each \(k\) corresponds to a specific position in the sequence. For instance, substituting \(k = -2\) gives the first term.
• Finite or Infinite: Sequences can be finite, like the one in our problem, which stops at \(k = 1\). It may also be infinite, continuing indefinitely.

Understanding sequences is essential, as they lay the groundwork for further study in areas like series, calculus, and analysis. They are the foundation upon which we can build to discover more complex mathematical relationships and patterns.