Question

Given the recursively defined formula a1=-3, a down n-1+4, determine which option is correct

Answer 1

It looks like you're working with a recursive sequence where the first term \( a_1 = -3 \) and it seems you might be indicating a recursive formula for subsequent terms. However, the formula you've provided appears to be incomplete or possibly misformatted: "a down n-1 + 4".

A common way to express a recursive relation is in the form:
\[ a_n = f(a_{n-1}) \]
where \( f \) is some function of the previous term.

If you're indicating that the next term is defined as:
\[ a_n = a_{n-1} + 4 \]
then we can proceed to find the terms of the sequence.

Given:
- \( a_1 = -3 \)

The subsequent terms would be calculated as follows:

- **For \( n = 2 \)**:
\[
a_2 = a_1 + 4 = -3 + 4 = 1
\]

- **For \( n = 3 \)**:
\[
a_3 = a_2 + 4 = 1 + 4 = 5
\]

- **For \( n = 4 \)**:
\[
a_4 = a_3 + 4 = 5 + 4 = 9
\]

- **For \( n = 5 \)**:
\[
a_5 = a_4 + 4 = 9 + 4 = 13
\]

Based on this pattern, the sequence appears to increase by 4 for each subsequent term. The general form of the \( n \)-th term in this case is:
\[
a_n = a_1 + (n - 1) \cdot 4
\]
Substituting \( a_1 = -3 \):
\[
a_n = -3 + (n - 1) \cdot 4
\]
This simplifies to:
\[
a_n = -3 + 4n - 4 = 4n - 7
\]

If you provide me with the specific options you're considering, I can help you determine which is correct based on this analysis!