Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write the indicated sum in sigma notation. $$ 1+2+3+\cdots+41 $$

Step-by-Step Solution

Answer

\[ \sum_{n=1}^{41} n \]

Step 1: Identify the Sequence Pattern

The given sequence is an arithmetic sequence, starting at 1 and ending at 41, with a common difference of 1 between consecutive terms.

Step 2: Define the General Term

For an arithmetic sequence starting at 1 with a common difference of 1, the general term can be given by \(a_n = n\), where \(n\) represents the position of the term in the sequence.

Step 3: Determine the Number of Terms

To find the number of terms, observe the first term is 1 and the last term is 41. Since it's an arithmetic sequence with a common difference of 1, the number of terms \(N\) is 41.

Step 4: Write the Sigma Notation

Using the general term from Step 2 and the number of terms from Step 3, the sum can be expressed in sigma notation as: \[ \sum_{n=1}^{41} n \] where the lower limit of summation is 1, and the upper limit is 41.

Key Concepts

Arithmetic Sequences

Arithmetic sequences are a type of sequence where each term after the first is formed by adding a constant difference to the previous term. This difference is known as the common difference. In the example provided, the sequence starts at 1 and progresses to 41 with a common difference of 1.

If the first term is denoted by \(a_1\), each subsequent term can be expressed in terms of the first term and the common difference \(d\):

• The second term \(a_2 = a_1 + d\)
• The third term \(a_3 = a_2 + d = a_1 + 2d\)
• The general term \(a_n = a_1 + (n-1)d\)

For this sequence, since \(a_1 = 1\) and \(d = 1\), the general term is simply \(a_n = n\). This simplicity makes arithmetic sequences easy to work with when analyzing the sequence's pattern and calculating sums. Understanding this structure can help in identifying and manipulating similar sequences in various mathematical problems.

Sigma Notation in Mathematics

Sigma notation is a compact way to represent a sum of terms in a sequence. The symbol \(\Sigma\) (capital Greek letter sigma) indicates that we should sum the terms following a specific formula.

To express a sum using sigma notation, identify the general term formula and the range of indices that the terms cover. In this problem, the sequence is expressed by the general term \(a_n = n\), and we are summing from the first term \(n=1\) to the 41st term \(n=41\).

The sigma notation for this sequence is:

• \( \sum_{n=1}^{41} n \)

Here, the expression after the sigma symbol, \(n\), is the term to be summed over the specified range. The numbers below and above the sigma indicate the starting and ending values for \(n\). This notation is not only concise but also provides a clear framework for calculating the total sum.

Sequences and Series

In mathematics, a sequence is an ordered list of numbers defined by some rule of progression. An arithmetic sequence is one such variety, characterized by a constant increment between consecutive terms. A series, on the other hand, involves summing the terms of a sequence.

When addressing sequences and series, understanding the distinction between the two is fundamental. While sequences simply list terms in order, series require you to add the terms together, which leads to a cumulative value.

For arithmetic sequences, the sum of the series can often be calculated using sigma notation or by employing formulas designed specifically for arithmetic series. Remember that in an arithmetic series, because of the uniform increment between terms, we can also use the formula:

• \( S = \frac{N}{2} (a_1 + a_N) \)

This formula provides a quick way to find the sum without individually adding each term, where \(N\) is the number of terms, \(a_1\) is the first term, and \(a_N\) is the last term of the series.

Summation Notation

Summation notation is a valuable tool in mathematics for representing the sum of a sequence of quantities. This is especially useful when dealing with long sequences where writing out all terms would be cumbersome. Summation uses the sigma symbol \(\Sigma\) accompanied by an index that indicates the terms to be added.

Understanding how to use summation notation effectively requires familiarity with the sequential structure it represents. In the context of our example, the sigma notation \(\sum_{n=1}^{41} n\) tells us to calculate the sum of the numbers from 1 through 41.

When using summation notation:

• The lower limit (e.g., \(n=1\)) indicates the starting index.
• The upper limit (e.g., \(n=41\)) indicates the ending index.
• The expression to the right of sigma specifies the general formula of the terms to be summed.

This organized format is integral in higher mathematics, where clarity and brevity are crucial, promoting efficient problem-solving and communication of ideas.