Problem 57 - Beginning and Intermediate Algebra

Question

Write each series using summation notation. 5+6+7+8+9+10

Step-by-Step Solution

Answer

The given series can be written in summation notation as: \[\sum_{i=0}^{5} (5 + i \cdot 1)\]

Step 1: Determine the first term (a) and the common difference (d)

Observe the given series: 5, 6, 7, 8, 9, 10. We can see that the first term (a) is 5, and the common difference (d) between each consecutive term is 1 (6-5=1, 7-6=1, and so on).

Step 2: Count the number of terms (n)

In the given series, there are six terms: 5, 6, 7, 8, 9, and 10. So, n=6.

Step 3: Write the series in summation notation

Now that we have the first term (a), the common difference (d), and the number of terms (n), we can write the series using summation notation. In general, an arithmetic series can be written as: \[\sum_{i=0}^{n-1} (a + i \cdot d) = a+(a+d)+(a+2d)+\cdots+(a+(n-1)d)\] In this case, a=5, d=1, and n=6. Therefore, the series can be written as: \[\sum_{i=0}^{5} (5 + i \cdot 1)\]

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference between terms is known as the "common difference." When you look at the series 5, 6, 7, 8, 9, and 10, you can identify it as arithmetic because each term increases by the same amount. Arithmetic series are important because they have straightforward patterns that are easy to predict and calculate. You can write any arithmetic series by determining a few key components, such as the first term and the common difference. This makes it simple to express the entire series in a compact, mathematical form.

Common Difference

The common difference is a crucial part of understanding arithmetic series. It tells you how much you add to go from one term to the next. In the series given:
- The first term is 5. - The second term is 6. By subtracting the first term from the second (6-5), we find that the common difference is 1.

• The same calculation holds for other terms: 7-6 = 1, 8-7 = 1.
• This consistent difference is what classifies the series as arithmetic.

Grasping the concept of the common difference is essential for writing and understanding the pattern in which numbers appear in an arithmetic sequence.

Series Representation

Representing a series is all about expressing it in a way that's efficient and easy to comprehend. With an arithmetic series, once you know the first term and the common difference, you can represent the entire series compactly.
For the series 5, 6, 7, 8, 9, 10: - Recognize the first term (5) - Identify the common difference (1) - Count the number of terms (6)
Then, we represent the series using a formula that incorporates these components, allowing us to write the series succinctly without having to list each term individually.

Mathematical Notation

Mathematical notation is a way to represent numbers and operations in a concise manner. In arithmetic series, summation notation is often used to represent the series efficiently. This is often expressed as a sigma (∑) followed by an expression that defines the terms of the series.
For example, the series 5, 6, 7, 8, 9, 10 can be written using summation notation as: \[ \sum_{i=0}^{5} (5 + i \cdot 1) \] This notation indicates that the series starts at 5 and continues by adding 1 each time, for a total of six terms. Summation notation is powerful because it can express long series in a simple formula, making calculations and analysis much easier.