Problem 31 - Algebra for College Students

Question

Write each series in summation notation. Use the index is and let i begin at I in each summation. $$x_{3}+x_{4}+x_{5}+\cdots+x_{50}$$

Step-by-Step Solution

Answer

\( \sum_{i=3}^{50} x_i \)

Step 1: Identify the Sequence

The given series is \( x_{3} + x_{4} + x_{5} + \cdots + x_{50} \). Identify the starting and ending terms of the series. Here, the series starts at \( x_{3} \) and ends at \( x_{50} \).

Step 2: Determine the Index Range

The index in each term of the series is shown as \( x_i \). The series begins with \( x_{3} \) and ends with \( x_{50} \). Therefore, the index \( i \) starts at 3 and goes to 50.

Step 3: Write in Summation Notation

Summation notation is written as \sum_{i=starting\ index}^{ending\ index} f(i) \. For this series, \( f(i) = x_i \), the starting index is 3, and the ending index is 50. Therefore, the summation notation is \sum_{i=3}^{50} x_i \.

Key Concepts

series

A series is the sum of the terms of a sequence. Imagine you have a list of numbers, called a sequence, and you add them up one after another. This sum is what we call a series. For example, in the series \( x_{3} + x_{4} + x_{5} + \cdots + x_{50} \), each term in the series is represented by \( x_i \). The process of summing these terms can be done in many ways, but summation notation makes this process very compact and clear.
The terms in a series can follow a specific pattern, and identifying this pattern is usually the first step in writing the series in summation notation.
Remember, a series doesn't just have to be about numbers; it can involve any mathematical expressions, as long as you are following the rules of addition and the given pattern.

index range

The index range is a key part of understanding summation notation. It tells us where to start and end the sum. In the series \( x_{3} + x_{4} + x_{5} + \cdots + x_{50} \), the index range specifies the first term and the last term. Here:

• The starting index is 3, as the series begins at \( x_{3} \).
• The ending index is 50, as the series ends at \( x_{50} \).

When we use summation notation, we place this starting and ending index within the summation symbol. The correct notation here is \sum_{i=3}^{50} x_i\.

• The starting index (3) is written at the bottom of the summation symbol.
• The ending index (50) is placed at the top of the symbol.

This structure helps in keeping track of how many terms you are summing and their positions in the sequence.

summation symbol

The summation symbol, \sum \, is used to denote the sum of a sequence of terms. It provides a compact way to write long sums, making equations easier to read and understand.
The general form of the summation notation is \sum_{i=start}^{end} f(i) \, where:

• \sum \ is the summation symbol.
• \{i=start\} tells us the index variable (i) starts at a particular value.
• \{end\} indicates the last value of the index in the sum.
• \(f(i) \) is the function or sequence formula.

So, for \sum_{i=3}^{50} x_i \:

• \sum \ initiates the summation.
• The index variable \{i=3\} starts at 3.
• The upper limit is 50.
• The function \(f(i) = x_i \) tells us what each term in the series looks like.

This method consolidates all the terms from the starting to the ending index into a single, manageable expression.