Problem 67 - Algebra and Trigonometry

Question

Express each sum using summation notation. \(a+(a+d)+(a+2 d)+\cdots+(a+n d)\)

Step-by-Step Solution

Answer

\(\sum_{k=0}^{n} (a + kd)\)

Step 1: Identify the pattern

Observe the given series: \(a + (a+d) + (a+2d) + \cdots + (a+nd)\). Notice that each term increases by a common difference \(d\). The general form of each term can be represented as \(a + kd\), where \(k\) is an integer.

Step 2: Establish the limits for the summation

Determine the range of the index \(k\). The series starts from \(k=0\) (where the term is \(a\)) and goes up to \(k=n\) (where the term is \(a+nd\)).

Step 3: Write the summation notation

Summation notation uses the symbol \(\sum\) to represent the sum of a sequence. Combine the general term and the limits identified:\[\sum_{k=0}^{n} (a + kd)\].

Key Concepts

Series

When we talk about a series, we're referring to the sum of the terms of a sequence. In your textbook exercise, you encounter a series like this: \(a + (a+d) + (a+2d) + \cdots + (a+nd)\). Each of these terms follows a specific pattern, forming a sequence. When you add up all these terms, it becomes what we call a series.
This series is called an arithmetic series because each term increases by the same amount, known as the common difference.
Understanding how to express series in different mathematical forms, like summation notation, is crucial. Summation notation allows us to write the series more compactly and is a powerful way to handle sums in algebra and calculus.

Common Difference

The common difference is the consistent amount by which each term in an arithmetic sequence increases. If you look at the series provided: \(a + (a+d) + (a+2d) + \cdots + (a+nd)\), you'll see that each term goes up by \(d\).
The first term is \(a\), the second term is \(a+d\), the third term is \(a+2d\), and so forth until the \((n+1)\)-th term, which is \(a+nd\).

Understanding the common difference is essential because it helps us identify the pattern in the series. This pattern is then used to write the series in summation notation. With the common difference defined, we can generalize the terms of the series and effectively sum them using summation notation.

Index

The index in summation notation specifies which term in the sequence you are considering. In the summation notation form, an index variable (like \(k\)) helps to indicate the position of each term in the series. In the given exercise solution, the series is expressed through summation notation as: \[\[\begin{equation} \sum_{k=0}^{n} (a + kd) \end{equation}\]\].
Here, \(k\) is the index that starts at 0 and goes up to \(n\).

- The lower limit of the index \(k=0\) indicates the first term \(a\).
- The upper limit \(k=n\) indicates the last term \(a+nd\).
The index thus helps us to know how many terms we are adding together and also gives us a compact way to represent the sequence in a mathematical form.
Manipulating the index correctly is key to accurately converting a series to and from summation notation.