Question

The first four terms of a sequence are given.

7, 11, 15, 19, ...

The pattern of the sequence continues in the same manner. Jenna creates a function ff to calculate the xx th term of the sequence where f, of, 1, equals, 7f(1)=7 .

Which function did Jenna create?

Answer 1

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.