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In an arithmetic sequence, the first term is the starting point of the sequence. This is typically denoted by the symbol \(a_1\). It is the number that appears first in the list of terms. To understand this better, let's look at the sequence given in the exercise: \[2.5, 3, 3.5, 4, \dots\]. In this case, the first term \(a_1\) is 2.5. The first term serves as the foundation upon which the rest of the sequence is built.
Knowing the first term is important, as it helps in finding any other term in the sequence using the formula for the nth term in an arithmetic sequence:

The common difference is a consistent value that separates the terms in an arithmetic sequence. It is denoted by the symbol \(d\). The common difference is fundamental because it shows how much each term increases or decreases as you move from one term to the next.
To find the common difference, you typically subtract the first term from the second term:
For the given sequence
\[2.5, 3, 3.5, 4, \dots\]
the common difference \(d\) can be calculated as follows:
\(d = a_2 - a_1\)
Here, \(a_2 = 3\) and \(a_1 = 2.5\). So, \(d = 3 - 2.5 = 0.5\).
The common difference 0.5 means that as you go from one term to the next in the sequence, you are adding 0.5. This keeps the sequence in a steady pattern.

A sequence is an ordered list of numbers. Each number in the sequence is called a term. In an arithmetic sequence, each term after the first is obtained by adding the common difference to the preceding term. The general form for an arithmetic sequence can be expressed as:
\[a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \dots\]
In our example, you can see this:
Starting with the first term 2.5 and adding the common difference 0.5, you get the next term:
\[2.5 + 0.5 = 3\]
Repeating this process for the next terms confirms the sequence:
\[3 + 0.5 = 3.5\]
\[3.5 + 0.5 = 4\]
Understanding how each term is generated helps you predict future terms and identify patterns in the sequence. This is particularly helpful in solving more complex problems involving sequences.