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An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is referred to as the common difference.
For example, in the sequence 1, 2, 3, 4, ..., 100, the common difference is 1.

• This means each term increases by 1 from the previous term.
• The sequence starts at 1 and ends at 100.

Understanding this pattern is crucial because it helps in finding any term in the sequence directly by using a formula.
In general, the nth term of an arithmetic sequence can be calculated using the formula:
\( a_n = a_1 + (n-1) imes d \)
where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term (which is 1 here), and
• \( d \) is the common difference (which is also 1).

This formula allows you to determine any term in the sequence without listing all the terms.

Series notation is a shorthand way to express the sum of the terms of a sequence. This is very useful for long sequences as it provides a concise way to represent their sum.
Sigma notation, denoted by the Greek letter \( \Sigma \), is commonly used to denote series.
For the example sequence 1, 2, 3, ..., 100, the sum can be written using sigma notation as follows:

• \( \sum_{n=1}^{100} n \)

This expression tells us to sum up all values of \( n \) starting from 1 and ending at 100.
The lower part of the sigma (below the \( \Sigma \)) indicates where the sequence starts, which is \( n=1 \) in this case, and the number above it shows where it ends, \( n=100 \).
Using sigma notation simplifies the representation of a series, making it easier to manipulate and understand.

The general term is a critical concept in understanding sequences and series. It describes each term in a sequence based on its position (n) and provides a formula to compute elements of the sequence without listing all previous terms.
In arithmetic sequences like our example, the general term is particularly straightforward.
For the sequence 1, 2, 3, ..., 100, the general term \( a_n \) is simply \( n \).
This means that the nth term equals \( n \) itself, i.e., the 50th term is 50, the 100th term is 100, and so forth.

• This simplicity is because the sequence increases by a consistent increment of 1.

Understanding the general term helps in tasks such as finding specific terms in the sequence directly or expressing the sequence in sigma notation.
The clarity of the general term aids not just in expressing sequences compactly but also in performing operations like finding their sums efficiently.