To determine the correct explicit formula for the given sequence \(3, 7, 11, 15, 19\), we first observe the pattern in the sequence.
The first term \(a_1 = 3\) corresponds to \(n = 1\). The second term \(a_2 = 7\) corresponds to \(n = 2\). The third term \(a_3 = 11\) corresponds to \(n = 3\). The fourth term \(a_4 = 15\) corresponds to \(n = 4\). The fifth term \(a_5 = 19\) corresponds to \(n = 5\).
Now, let's look for a pattern in the terms:
- The first term is \(3\).
- The difference between consecutive terms is constant: \[ 7 - 3 = 4, \] \[ 11 - 7 = 4, \] \[ 15 - 11 = 4, \] \[ 19 - 15 = 4. \]
Since the difference is constant, the sequence is arithmetic with a common difference of \(4\).
Next, we can use the formula for the \(n\)-th term of an arithmetic sequence: \[ a_n = a_1 + (n-1)d, \] where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
Plugging in \(a_1 = 3\) and \(d = 4\): \[ a_n = 3 + (n - 1) \cdot 4. \] Simplifying this: \[ a_n = 3 + 4n - 4 = 4n - 1. \]
Now let's analyze the options given:
- Option #1: \(a_n = 3 + 4n\) (not correct; substituting \(n=1\) gives \(7\) instead of \(3\))
- Option #2: \(a_n = 4 + 3n\) (not correct; substituting \(n=1\) gives \(7\) instead of \(3\))
- Option #3: \(a_n = -1 + 4n\) (correct; substituting \(n=1\) gives \(3\))
Thus, the correct subscript notation for the explicit formula is:
Option #3: \(a_n = -1 + 4n\).