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!