Problem 30 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{6}(3 i-2)$$

Step-by-Step Solution

Answer

The sum of the series is 51.

Step 1: Understand the Series Formula

The given series is \( \sum_{i=1}^{6}(3i - 2) \). This summation notation means that you have to calculate the expression \(3i - 2\) for each integer value from \(i = 1\) to \(i = 6\) and then add up all these values to get the total sum of the series.

Step 2: Evaluate the Expression for Each Term

Calculate each term of the series individually by substituting the integers from 1 to 6 into the expression \(3i - 2\).- For \(i = 1\), the term is \(3(1) - 2 = 3 - 2 = 1\).- For \(i = 2\), the term is \(3(2) - 2 = 6 - 2 = 4\).- For \(i = 3\), the term is \(3(3) - 2 = 9 - 2 = 7\).- For \(i = 4\), the term is \(3(4) - 2 = 12 - 2 = 10\).- For \(i = 5\), the term is \(3(5) - 2 = 15 - 2 = 13\).- For \(i = 6\), the term is \(3(6) - 2 = 18 - 2 = 16\).

Step 3: Sum the Calculated Terms

Add up all the terms that were calculated in the previous step:\[1 + 4 + 7 + 10 + 13 + 16\]Perform the addition step-by-step:1. \(1 + 4 = 5\)2. \(5 + 7 = 12\)3. \(12 + 10 = 22\)4. \(22 + 13 = 35\)5. \(35 + 16 = 51\)

Step 4: Conclusion

The sum of the series \( \sum_{i=1}^{6}(3i - 2) \) is therefore \(51\).

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. This constant difference is known as the common difference. For example, in the sequence 2, 5, 8, 11, 14, each term increases by 3, making 3 the common difference.
In any arithmetic series, the key components are:

• First term (\(a_1\)
• Common difference (\(d\)
• Number of terms (\(n\)

To find the sum of an arithmetic series, you can use the formula:\[S_n = \frac{n}{2} (2a_1 + (n-1) \cdot d)\]\This tells you how to calculate the sum based on the first term, the number of terms, and the common difference. In our exercise, however, we can simplify by evaluating each term of the series individually and adding them up, as outlined in the problem's solution.

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. The symbol used for summation is the Greek letter sigma (\(\Sigma\)). This notation visibly compiles a series into a more compact form, often expressed as:\[\sum_{i=a}^{b} f(i)\]Here,

• \(i\) represents the index of summation, which takes on integer values.
• \(a\) is the starting index.
• \(b\) is the ending index.
• \(f(i)\) is the expression for each term, typically a function of \(i\).

In our example, the summation is \(\sum_{i=1}^{6}(3i - 2)\), meaning we calculate \(3i - 2\) for each integer \(i\) from 1 to 6. This approach simplifies the process of summing many terms by outlining only the essential components of the series.

Mathematical Expression Evaluation

Mathematical expression evaluation involves calculating the value of an expression by applying arithmetic operations. Each part of the expression is substituted with specific values to perform the calculations step-by-step.
Here's how you handle it in our exercise:

• Identify the expression to be evaluated: \(3i - 2\).
• Determine each value of \(i\) within the range specified by the summation (from 1 to 6).
• Substitute \(i\) into the expression to find each specific term:
• For \(i = 1\), calculate \(3 \cdot 1 - 2 = 1\).
• For \(i = 2\), calculate \(3 \cdot 2 - 2 = 4\).
• Continue this until \(i = 6\).
• Finally, sum all the calculated terms.

This systematic approach ensures every part of the calculation is correct, leading to an accurate final sum for the series.