Question

Given the recursively defined formula a = -3, aN = aN-1 + 4, determine which option is the correct function notation.

Answer 1

To analyze the recursively defined sequence given by \( a_1 = -3 \) and \( a_n = a_{n-1} + 4 \), we can determine the explicit formula for \( a_n \).

1. Start with the recursive definition:
- \( a_1 = -3 \)
- \( a_2 = a_1 + 4 = -3 + 4 = 1 \)
- \( a_3 = a_2 + 4 = 1 + 4 = 5 \)
- \( a_4 = a_3 + 4 = 5 + 4 = 9 \)
- \( a_5 = a_4 + 4 = 9 + 4 = 13 \)

From these calculations, we can observe the pattern in the sequence:
- \( a_1 = -3 \)
- \( a_2 = 1 \)
- \( a_3 = 5 \)
- \( a_4 = 9 \)
- \( a_5 = 13 \)

Notice that the sequence is increasing by 4 each time.

2. We can determine the explicit formula for \( a_n \) by recognizing that this sequence resembles an arithmetic sequence, where the first term is \( -3 \) and the common difference is \( 4 \).

The formula for the \( n \)-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n-1)d
\]
Where \( a_1 \) is the first term and \( d \) is the common difference.

Plugging our values into this formula:
- \( a_1 = -3 \)
- \( d = 4 \)

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

3. Simplifying this expression:
\[
a_n = -3 + 4n - 4
\]
\[
a_n = 4n - 7
\]

Thus, the correct function notation for the recursively defined sequence is:
\[
\boxed{a_n = 4n - 7}
\]