Problem 30 - Algebra and Trigonometry

Question

find each indicated sum. $$ \sum_{i=1}^{6} 7 i $$

Step-by-Step Solution

Answer

The sum of the series presented in the exercise is 147.

Step 1: Identifying the Series

The sum can be written as \(7 \cdot 1 + 7 \cdot 2 + ... + 7 \cdot 6 \). This indicates that we are summing the values from 1 to 6 after each has been multiplied by 7.

Step 2: Performing the Calculation

By performing each multiplication and then adding the results we get: \(7 \cdot 1 + 7 \cdot 2 + 7 \cdot 3 + 7 \cdot 4 + 7 \cdot 5 + 7 \cdot 6 = 7 + 14 + 21 + 28 + 35 + 42 = 147 \).

Step 3: Sum of the Series

The sum of the series \(\sum_{i=1}^{6} 7 i \) is calculated as 147.

Key Concepts

Arithmetic Series

When you see the term 'arithmetic series', think about a sequence of numbers, where each term after the first is generated by adding a constant to the previous term. This constant is called the 'common difference'. For example, in the sequence 2, 5, 8, 11, 14, each number increases by 3. Therefore, 3 is the common difference. In a series, you'll simply add all these terms together.

Let's consider an arithmetic series in a generalized form: the first term is denoted by 'a', and 'd' is the common difference. If you want to find the sum of the first 'n' terms, there's a handy formula: \( S_n = \frac{n}{2} (2a + (n-1)d) \). This formula saves you from adding each term manually.

• Understand the formula structure, where \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference.
• Using the example above, with a first term of 2 and a common difference of 3, you can quickly apply the formula to find the sum of the terms.

Mathematical Series

A mathematical series is essentially the sum of a sequence of numbers. Sequences can follow different patterns, like arithmetic, geometric, or other finite and infinite forms. Understanding the nature of the sequence is key to finding the series sum. Each series type has its own approach, and formulas make it easier to handle more complex sums.

In an arithmetic series, we discussed that the terms increase by a consistent amount, known as the common difference. Geometric series, on the other hand, multiply each term by a constant to get to the next one. These patterns structure the series and influence how you calculate the sum.

When working with mathematical series, consider:

• Whether the series is finite or infinite.
• If the series follows a particular pattern, like arithmetic or geometric, which defines the sum formula to use.
• An understanding that some series can be simply written in summation notation \( \sum \), which provides a compact way to express the series.

Sum of Series

The sum of a series is what you get when you add all the terms in a sequence together. Summation notation (\( \sum \)) provides a tidy way to express this process. For example, the notation \( \sum_{i=1}^{n} a_i \) reads as "the sum of \(a_i\) from \(i=1\) to \(n\)." Here, \(a_i\) represents the terms you are summing.

In the exercise provided, we determined the sum of the series \( \sum_{i=1}^{6} 7i \). This means that you multiply each integer from 1 to 6 by 7 and add those products together. Each iteration adds one more step to the sum until the final number in the series is included.

Key takeaways for solving the sum of a series:

• Identify the pattern or rule of the sequence.
• Use summation notation to express the process compactly when needed.
• Apply formulas, like those used for arithmetic or geometric series, when they simplify the calculation process.
• Checking your calculations step-by-step ensures accuracy when computing manually.

Understanding these concepts will help you approach any series with confidence, using structured methods to find sums efficiently.