Problem 60 - A Graphical Approach to College Algebra

Question

Evaluate each sum. $$\sum_{i=1}^{5}(i-8)$$

Step-by-Step Solution

Answer

The sum is -25.

Step 1: Understanding the Summation

The given expression is a summation from 1 to 5 of the term \(i - 8\). This means you need to substitute each integer value from 1 to 5, subtract 8 from it, and then sum all these results.

Step 2: Calculating Each Term in the Series

Begin by substituting each value from 1 to 5 into the expression \(i - 8\) to find each term: - For \(i = 1\), \(1 - 8 = -7\)- For \(i = 2\), \(2 - 8 = -6\)- For \(i = 3\), \(3 - 8 = -5\)- For \(i = 4\), \(4 - 8 = -4\)- For \(i = 5\), \(5 - 8 = -3\)

Step 3: Summing the Calculated Terms

Now, add all the individual terms obtained in the previous step:\(-7 + (-6) + (-5) + (-4) + (-3) = -7 - 6 - 5 - 4 - 3 = -25\).

Step 4: Final Result

The sum of the series \(\sum_{i=1}^{5}(i-8)\) is \(-25\).

Key Concepts

Algebra

Algebra is like a puzzle. It helps us solve problems by creating equations or expressions. This involves finding unknown values by using different operations such as addition, subtraction, multiplication, and division. Think of algebra as building blocks for understanding how numbers relate to each other. In the context of the summation problem, we focus on the expression \(i - 8\). Here, \(i\) represents each number from 1 to 5. The expression shows how subtraction is used within algebra to transform one algebraic statement into another, making it simpler to work with.

• Variable: A symbol, like \(i\), that represents a number.
• Expression: A combination of variables, numbers, and operations, such as \(i - 8\).
• Substitution: Replacing a variable with a specific number.

Arithmetic Series

An arithmetic series is a sum of terms in a sequence where each term after the first is obtained by adding a constant difference to the previous term. In the given summation problem, the arithmetic series results from repeatedly applying the expression \(i - 8\) from \(i = 1\) to \(i = 5\). This means taking each step in the sequence, subtracting 8, and then adding up these new values.

• Terms: The individual numbers in the sequence, like \(-7, -6, -5, -4, -3\).
• Common Difference: Although each term is not acquired by adding a fixed number, subtracting 8 from each \(i\) creates a predictable pattern.
• Summation Notation: Uses the Greek letter \(\Sigma\) to signify adding up a range of terms.

Mathematical Expression Evaluation

Evaluating a mathematical expression involves calculating its value using given operations and numbers. For the expression \(\sum_{i=1}^{5}(i-8)\), we substitute each \(i\) value with integers from 1 to 5. This step-by-step substitution followed by subtraction gives us each term.
Once all terms \(-7, -6, -5, -4, -3\) are obtained, we add them together to find the total sum, which in this case, equals \(-25\).

• Substitute Values: Insert each specific integer into the expression.
• Perform Operations: Conduct the arithmetic for each term and the total sum.
• Summation Result: The end value after completing all calculations, simplified to a single number.