Problem 9 - Calculus

Question

In Problems 9-14, write the indicated sum in sigma notation. \(1+2+3+\cdots+41\)

Step-by-Step Solution

Answer

\(\sum_{i=1}^{41} i\) is the sigma notation.

Step 1: Understand the Series

The series given is an arithmetic series starting at 1 and ending at 41. It follows the form: 1, 2, 3, ..., 41 which is an arithmetic sequence.

Step 2: Identify the General Term

The general term of an arithmetic sequence is given by the formula \(a_i = a_1 + (i-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference. For this series, the first term \(a_1 = 1\) and the common difference \(d = 1\). Thus the general term \(a_i = i\).

Step 3: Determine the Last Term

Since the last term of the series is 41, we can determine the number of terms \(n\). This is already given by the last term when the sequence starts from 1 and has a difference of 1. Hence, \(n=41\).

Step 4: Write in Sigma Notation

To write the series in sigma notation, we use the general term and the limits of summation. The sigma notation for the sum \(1 + 2 + 3 + ... + 41\) is \(\sum_{i=1}^{41} i\).

Key Concepts

Arithmetic Series

An arithmetic series is a sum of terms of an arithmetic sequence. In simpler terms, it is when you add up numbers that have a constant difference between them. Consider the series 1, 2, 3, ..., 41 — this is an arithmetic series because each number increases by 1.
An important feature of arithmetic series is that they have a predictable pattern. This can make them easier to sum using specific formulas or notation. For instance, you can use the formula for the sum of an arithmetic series if you know the first term, the last term, and the number of terms.

Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This difference is known as the "common difference." For the sequence 1, 2, 3, ..., 41, each subsequent number increases by 1.

• The first term of this sequence is 1.
• The common difference is 1, since 2 - 1 = 1.
• Each term can be written as the one before it plus the common difference.

Understanding this concept is essential, as it helps you recognize which numbers belong to the sequence and allows you to easily predict future terms.

General Term Formula

The general term formula of an arithmetic sequence allows you to find any term in the sequence without listing all previous terms. The formula is given by:

• \( a_i = a_1 + (i-1) \cdot d \)

Where:

• \( a_i \) is the term you want to find.
• \( a_1 \) is the first term.
• \( i \) is the term number, or position, in the sequence.
• \( d \) is the common difference.

For the sequence 1, 2, 3, ..., 41, the general term is simply the sequence's position \( i \). So, \( a_i = i \). This means if you want the 3rd term, it's 3, the 20th term is 20, and so on. This straightforward relationship makes arithmetic sequences particularly easy to work with when compared to other types of sequences.

Limits of Summation

When writing an arithmetic series in sigma notation, it's important to define the limits of summation. These limits tell you where to start and stop your summation.
In our example, the series 1 + 2 + 3 + ... + 41 is written in sigma notation as:

• \( \sum_{i=1}^{41} i \)

Here's how the limits of summation work:

• The lower limit, \( i = 1 \), indicates the starting point of the series.
• The upper limit, \( i = 41 \), tells you where to end the series.
• \( i \) is the index of summation, which helps in calculating the general term for each step of the sum.

By specifying these limits, you compactly express what could be a long series of additions, saving time and effort. Understanding the limits of summation makes it much easier to transform an arithmetic sequence into a concise expression using sigma notation.