The given sequence is:
7, 11, 15, 19, ...
To analyze this sequence, we can observe the pattern:
- The first term is 7 (when \( n = 1 \))
- The second term is 11 (when \( n = 2 \))
- The third term is 15 (when \( n = 3 \))
- The fourth term is 19 (when \( n = 4 \))
To find the pattern, we can look at the differences between consecutive terms:
- \( 11 - 7 = 4 \)
- \( 15 - 11 = 4 \)
- \( 19 - 15 = 4 \)
Each term increases by 4, indicating that this is an arithmetic sequence with a common difference of 4.
The general formula for the \( n \)-th term of an arithmetic sequence is given by:
\[ a_n = a_1 + (n - 1) \cdot d \]
where \( a_1 \) is the first term and \( d \) is the common difference. Here, \( a_1 = 7 \) and \( d = 4 \).
Plugging these values into the formula:
\[ a_n = 7 + (n - 1) \cdot 4 \]
Simplifying this expression:
\[ a_n = 7 + 4n - 4 \]
\[ a_n = 4n + 3 \]
Thus, the function \( f(n) \) that Jenna created to calculate the \( n \)-th term of the sequence is:
\[ f(n) = 4n + 3 \]
This is the final answer.