Question

Write out the sums. (You do not need to evaluate them.) \(\sum_{j=1}^{6} 5(j-3)\)

Short answer

Based on the step by step solution provided, write the sum for the expression \(\sum_{j=1}^{6} 5(j-3)\) without evaluating it. Answer: \(\sum_{j=1}^{6} 5(j-3) = 5(-2) + 5(-1) + 5(0) + 5(1) + 5(2) + 5(3)\)

Step-by-step solution

Understand the expression

The given expression is \(\sum_{j=1}^{6} 5(j-3)\). This is a summation notation which means that we need to substitute the values of j, from 1 to 6, into the expression and add the terms together.

Write out the terms

Now, we will substitute the values of j from 1 to 6 into the expression and write out the terms without evaluating them. When \(j = 1\): \(5(1-3) = 5(-2)\) When \(j = 2\): \(5(2-3) = 5(-1)\) When \(j = 3\): \(5(3-3) = 5(0)\) When \(j = 4\): \(5(4-3) = 5(1)\) When \(j = 5\): \(5(5-3) = 5(2)\) When \(j = 6\): \(5(6-3) = 5(3)\)

Write out the sum

Finally, we write out the sum of all the terms without evaluating: \(\sum_{j=1}^{6} 5(j-3) = 5(-2) + 5(-1) + 5(0) + 5(1) + 5(2) + 5(3)\)

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators. In our exercise, we see the expression \(5(j-3)\), where \(j\) is the variable representing different numbers, and it is being multiplied by 5. The expression describes an arithmetic operation involving these components.

Algebraic expressions are essential in mathematics because they allow us to represent and manipulate numbers and quantities in a compact form. The variables in these expressions can take on various values, which makes algebra both a versatile and powerful tool.

When working with algebraic expressions, it's important to identify the parts of the expression correctly. This helps in simplifying or transforming the expression as needed. Simplification involves combining like terms and using operations correctly. It’s a fundamental skill needed when handling more complex algebra problems.

Arithmetic Sequences

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. In the context of our problem, the expression \(5(j-3)\) evaluates at different values of \(j\) to create a sequence. This specific sequence forms part of our summation problem when \(j\) is replaced with values from 1 to 6.

The difference between terms in an arithmetic sequence is called the common difference. It allows you to predict the next term in the sequence if given any term and the common difference. In our example, the substitution creates components like \(5(-2), 5(-1),\) and so on, aligned in a manner reflective of arithmetic progression.

Recognizing and understanding arithmetic sequences is crucial because they frequently occur in various areas of mathematics. They help in analyzing patterns and can be used to model real-world problems!

Mathematical Summation

Mathematical summation is the process of adding a sequence of numbers together. This concept is represented through summation notation, often expressed using the Greek letter sigma (\(\Sigma\)). In our exercise, \(\sum_{j=1}^{6} 5(j-3)\) instructs us to sum the outputs of \(5(j-3)\) as \(j\) varies from 1 through 6.

This notation simplifies difficult arithmetic and algebraic operations by compactly representing repeated addition of terms. Each term is generated by evaluating the expression \(5(j-3)\) for every integer value of \(j\) within the specified limits. While we don't evaluate the sum in this exercise, the notation uniquely specifies how each term adds up to form the series.

The power of summation extends beyond basic arithmetic sequences. It appears in calculus, statistics, and various complex mathematical models, offering a structured approach to summing intervals and functions easily.