The given sequence is: 7, 11, 15, 19.
To define the sequence explicitly, we can observe that it is an arithmetic sequence where each term after the first is generated by adding a fixed number (the common difference) to the previous term.
In this case, the common difference is:
11 - 7 = 4,
15 - 11 = 4,
19 - 15 = 4.
The first term \( a_1 \) is 7, and the common difference \( d \) is 4.
The general formula for the \( n \)-th term \( a_n \) of an arithmetic sequence can be written as:
\[ a_n = a_1 + (n - 1) \cdot d \]
Substituting \( a_1 = 7 \) and \( d = 4 \):
\[ a_n = 7 + (n - 1) \cdot 4 \]
Simplifying this expression gives:
\[ a_n = 7 + 4n - 4 = 4n + 3 \]
Thus, the explicit definition of the sequence is:
\[ a_n = 4n + 3 \]
for \( n = 1, 2, 3, \ldots \)
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