Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write the indicated sum in sigma notation. $$ 2+4+6+8+\cdots+50 $$

Step-by-Step Solution

Answer

\( \sum_{i=1}^{25} 2i \).

Step 1: Identify the Pattern

First, analyze the sequence to identify any patterns. The sequence given is 2, 4, 6, 8, ..., 50. Notice that each term increases by 2, which suggests an arithmetic sequence.

Step 2: Establish the General Term

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

Step 3: Determine Number of Terms

To find the number of terms in the sequence, solve for \( n \) when the general term is equal to the last term, which is 50. Set \( 2n = 50 \) and solve: \( n = 25 \). Therefore, the sequence has 25 terms.

Step 4: Write in Sigma Notation

The sigma notation for the sum of an arithmetic sequence is \( \sum_{i=1}^{n} a_i \). With \( a_i = 2i \) and \( n = 25 \), the expression becomes \( \sum_{i=1}^{25} 2i \). Thus, the sum in sigma notation is \( \sum_{i=1}^{25} 2i \).

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This difference is known as the 'common difference.' In our example, the sequence is 2, 4, 6, 8, ..., 50. Here:

• The first term, denoted as \(a_1\), is 2.
• The common difference, denoted as \(d\), is 2 (since 4 - 2 = 2, and similarly for the other terms).

For any arithmetic sequence, the formula to describe the sequence is\[ a_n = a_1 + (n-1) \times d \]
This formula allows us to find any term in the sequence if we know the first term and the common difference. This simple yet powerful technique is crucial for working with arithmetic sequences.
The systematic nature of arithmetic sequences makes it easy to calculate sums and other properties using basic formulas and gradual progression.

Decoding the General Term

The 'general term' of an arithmetic sequence, which is sometimes called the 'nth term,' allows us to calculate any specific term within the sequence. In the provided sequence, we identified:

• \( a_1 = 2 \) (the first term),
• \( d = 2 \) (the common difference).

Thus, substituting these into the general formula:\[ a_n = 2 + (n-1) \times 2 = 2n \]
The use of \(2n\) confirms that each term doubles from the previous integer position. For instance, if \( n = 3 \), we plug it into the equation to get:

• \( a_3 = 2 \times 3 = 6 \)

This confirms our sequence. The general term is not just for finding specific elements but for understanding the broader pattern in sequences that follow a predictable increase or decrease.
Such clarity in arithmetic sequences makes more complex operations like sums straightforward.

Determining the Number of Terms

Knowing how many terms are in a sequence is essential when performing operations like summing. In our sequence, identified by the general term formula \( 2n \), we set \(2n\) equal to the last term, 50, to solve for \(n\). This is done as follows:

• \( 2n = 50 \)
• Solve for \(n\): \( n = \frac{50}{2} = 25 \)

Thus, there are 25 terms in this sequence. Understanding this process is vital because correct application of formulas and logic ensures accurate calculation of sequences in mathematics.
The number of terms, along with the common difference and general term, allows for seamless conversion of a sequence to sigma notation, such as expressed in the sum \( \sum_{i=1}^{25} 2i \). Knowing the number of terms solidifies our ability to handle arithmetic sequences confidently.