Question

Given the recursively defined formula a1=−3 , an=an−1+4 , determine which option is the correct function notation.(1 point)
Responses

Answer 1

To find the function notation for the recursively defined sequence given by \( a_1 = -3 \) and \( a_n = a_{n-1} + 4 \), we can first compute the first few terms to identify the pattern.

1. \( a_1 = -3 \)
2. \( a_2 = a_1 + 4 = -3 + 4 = 1 \)
3. \( a_3 = a_2 + 4 = 1 + 4 = 5 \)
4. \( a_4 = a_3 + 4 = 5 + 4 = 9 \)
5. \( a_5 = a_4 + 4 = 9 + 4 = 13 \)

The sequence appears to be an arithmetic sequence with a first term of \(-3\) and a common difference of \(4\).

The general form of an arithmetic sequence can be expressed as:

\[
a_n = a_1 + (n - 1)d
\]

where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

Substituting our values where \( a_1 = -3 \) and \( d = 4 \):

\[
a_n = -3 + (n - 1) \cdot 4
\]

Simplifying that equation:

\[
a_n = -3 + 4n - 4 = 4n - 7
\]

Thus, the function notation for the given sequence would be:

\[
a_n = 4n - 7
\]

Now, please provide the multiple-choice options so that we can identify which one matches our derived function notation.