Problem 14 - University Calculus: Early Transcendentals

Question

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$2+4+6+8+10$$

Step-by-Step Solution

Answer

The sum is \( \sum_{n=1}^{5} 2n \).

Step 1: Identify the Pattern in the Sequence

The sequence given is 2, 4, 6, 8, 10. Notice that each term increases by 2. This is an arithmetic sequence with the common difference of 2.

Step 2: Express the General Term

For an arithmetic sequence, the general term can be expressed as \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term and \( d \) is the common difference. Here, \( a_1 = 2 \) and \( d = 2 \), so \( a_n = 2 + (n-1) imes 2 = 2n \).

Step 3: Determine the Number of Terms

Count the terms in the sequence: 2, 4, 6, 8, 10 are 5 terms. Thus, the sequence spans from \( n = 1 \) to \( n = 5 \).

Step 4: Express the Series in Sigma Notation

Using the general term \( 2n \) and the limits from \( n = 1 \) to \( n = 5 \), we can express the sum in sigma notation as \( \sum_{n=1}^{5} 2n \).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed number, known as the common difference, to the previous term. This type of sequence is straightforward and predictable, making it ideal for understanding basic sequence patterns.

• The sequence in the example is 2, 4, 6, 8, 10.
• Each number increases by 2 from the previous one.
• This consistent step of 2 is what identifies it as an arithmetic sequence.

This simple, steady increase or decrease is the hallmark of arithmetic sequences. They follow a straightforward rule that makes their analysis and representation much easier compared to more complex sequences.

Common Difference

The common difference is a key component of an arithmetic sequence. It is the fixed number that you add (or subtract) to get from one term to the next. Understanding the common difference allows you to predict future terms in the sequence.

• In our example sequence 2, 4, 6, 8, 10, the common difference is 2.
• This means each term increases by 2 as you move down the sequence.

To find the common difference, simply subtract any term from the one that follows it. The consistent result across the sequence confirms it's arithmetic. This difference simplifies the process of defining the sequence's general term and constructing its series representation.

Series Representation

When working with sequences, especially arithmetic sequences, representing them as a series is a powerful tool. A series is essentially the sum of the terms of a sequence.In the example, the series representation involves summing the arithmetic sequence 2, 4, 6, 8, 10. Such representations help in finding the total or cumulative value of the sequence terms.

• We use sigma notation to express this sum: \( \sum_{n=1}^{5} 2n \).
• This notation indicates that you are summing terms from the first to the fifth.

This series representation is compact and efficient, showing both the sequence pattern and the sum process without listing all terms, making it very useful for mathematical analysis and computation.

General Term

The general term of a sequence provides a formula to find any term in the sequence without listing all preceding terms. For an arithmetic sequence, the general term is given by:\[ a_n = a_1 + (n-1) \times d \]where:

• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.

In our example, the first term \(a_1\) is 2, and the common difference \(d\) is 2. Therefore, the general term becomes \(a_n = 2 + (n-1) \times 2 = 2n \). This formula lets you find any term directly, such as the 10th or 100th term.
Using the general term provides a clear, formulaic approach to understanding the sequence, thus avoiding the need to manually list each term.