The given arithmetic sequence is represented by the formula:
- an = 2 + 5n + 1
We can simplify this formula:
- an = 5n + 3
This equation describes an arithmetic sequence where:
- The first term can be calculated by substituting the first value of n (which is typically 0 in sequences):
a0 = 5(0) + 3 = 3
- The common difference (d) of the sequence can be identified as 5 from the term coefficient of n.
In terms of the domain for n, since n can take any integer value (if we are considering integer sequences) or any real number (if we’re not limiting n), we can express the domain as follows:
- For integers:
n ∈ ℤ(which means any integer, including negative, zero, and positive integers) - For real numbers:
n ∈ ℝ(which includes all real numbers)
To summarize, the domain for n in the arithmetic sequence defined by an = 2 + 5n + 1 can either be:
- n is an integer:
n ∈ ℤ - n is a real number:
n ∈ ℝ
In most cases, especially in arithmetic sequences typically taught in school, the domain is often limited to integers for practical applications.