Problem 59 - Precalculus: Mathematical for Calculus

Question

Write the sum using sigma notation. $$1+2+3+4+\dots+100$$

Step-by-Step Solution

Answer

\(\sum_{n=1}^{100} n\)

Step 1: Identify the Sequence

The sequence given is an arithmetic sequence: \(1, 2, 3, \ldots, 100\). It starts at 1 and continues up to 100, increasing by 1 each step.

Step 2: Express the General Term

In an arithmetic sequence, the general term \(a_n\) can be expressed as \(a_n = a_1 + (n-1) \cdot d\), where \(d\) is the common difference. For our sequence, \(a_1 = 1\) and \(d = 1\), so the \(n\)-th term is \(a_n = n\).

Step 3: Write the Index Range

The sequence begins with \(n = 1\) and ends with \(n = 100\), so the index range is from 1 to 100.

Step 4: Use Sigma Notation

The sum of the sequence can be written in sigma notation as: \[\sum_{n=1}^{100} n\] This represents the sum of all integers from 1 to 100.

Key Concepts

Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which each term after the first is obtained by adding a fixed, constant number, called the common difference, to the previous term. This constant difference forms the arithmetic backbone of the sequence. In our example, the sequence is:

• 1, 2, 3, ..., 100

Here, the common difference (\(d\)) is 1, as each term increases by 1 compared to the previous one. Arithmetic sequences are fundamental in mathematics, particularly when summing a series of consecutive numbers.

General Term

Understanding the general term in an arithmetic sequence allows us to define any term within the sequence without having to list all previous terms. The formula for the general term \(a_n\) of an arithmetic sequence is:\[a_n = a_1 + (n-1) \cdot d\]where:

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

In the provided sequence, \(a_1 = 1\) and \(d = 1\) are given. Therefore, the general term simplifies to \(a_n = n\), indicating that the \(n\)-th term is directly equal to \(n\). This makes this particular sequence quite straightforward.

Index Range

The index range of a sequence specifies which terms are included in the sequence. For an arithmetic sequence, the index range is determined by the position of the terms with respect to \(n\). In the example at hand:

• The sequence begins at \(n = 1\).
• It ends at \(n = 100\).

This tells us that we are examining every integer from 1 through 100. In sigma notation, this is represented as \(\sum_{n=1}^{100}\), clearly indicating that our sum starts at \(n = 1\) and concludes with \(n = 100\). This range provides a concise way to express the scope of the sequence or sum being considered.