Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of each series. $$\sum_{i=1}^{12}(-5-8 i)$$

Step-by-Step Solution

Answer

The sum of the series is -684.

Step 1: Identify the Formula

The given series is \ \(\sum_{i=1}^{12}(-5-8i)\). This is an arithmetic series. We will use the formula for the sum of an arithmetic series: \\[ S_n = \frac{n}{2} (a_1 + a_n) \] \where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.

Step 2: Determine the Number of Terms

In this series, \(n = 12\) because the summation runs from \(i = 1\) to \(i = 12\).

Step 3: Calculate the First Term

The first term in the series, \(a_1\), occurs when \(i = 1\). Substitute \(i = 1\) into the expression: \\[ a_1 = -5 - 8 \cdot 1 = -13 \]

Step 4: Calculate the Last Term

The last term in the series, \(a_n\), occurs when \(i = 12\). Substitute \(i = 12\) into the expression: \\[ a_n = -5 - 8 \cdot 12 = -101 \]

Step 5: Substitute Values into the Formula

Now substitute the values of \(n\), \(a_1\), and \(a_n\) into the arithmetic series sum formula: \\[ S_{12} = \frac{12}{2} (-13 + (-101)) \] \\[ S_{12} = 6 \times (-114) \]

Step 6: Compute the Result

Multiply the terms inside the brackets: \\[ S_{12} = 6 \times (-114) = -684 \]

Step 7: Conclusion

The sum of the series \(\sum_{i=1}^{12}(-5-8i)\) is \(-684\).

Key Concepts

Sum of Series

When we talk about the 'Sum of Series,' we are referring to the total sum that results from adding up all the terms in a sequence. In the context of an arithmetic series, as in our example, the terms have a specific pattern, which makes calculating the sum straightforward once we know the series is arithmetic.
To find the sum of an arithmetic series, we use a special formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \] where:

• \( S_n \) is the sum of the series,
• \( n \) is the number of terms,
• \( a_1 \) is the first term, and
• \( a_n \) is the last term.

In our sample exercise, the terms follow a linear rule, and we use this clearly defined formula to find the total sum of all the terms from the first to the last term in sequence. This approach makes it much faster than adding each term individually.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between each consecutive term is constant. This difference is known as the "common difference." Understanding this concept is crucial when dealing with series, especially arithmetic series.
To identify an arithmetic sequence, simply check if the difference between terms is the same consistently. For example, with the sequence \(-5, -13, -21, ..., -101\), we subtract \(8\) each time to find the next term. This subtraction of \(8\) is the common difference in our case.
The knowledge of an arithmetic sequence helps us to find the first term \(a_1\) and the last term \(a_n\) easily, which are essential for calculating the sum of the series. Such sequences appear frequently in mathematical problems, so understanding their properties and how to compute them is very useful.

Mathematical Formula

Mathematical Formulas are used to provide concise and precise methods for solving complex problems. In the context of arithmetic series, formulas simplify finding solutions to problems that would be cumbersome to do manually.
The sum formula for an arithmetic series \( S_n = \frac{n}{2} (a_1 + a_n) \) is a small snippet of mathematical power. It condenses what could be many steps of arithmetic into one elegant operation. Using this formula requires knowing a few key pieces of information:

• The number of terms \( n \),
• The first term \( a_1 \),
• The last term \( a_n \).

These values help you to employ the formula swiftly to obtain the series sum. Formulas act as tools that transform complex sequences of calculations into a manageable and efficient process. Once armed with the formula, solving similar problems becomes routine and less susceptible to error.