The given sequence is 11, 8, 5, 2. To define it recursively, we need to establish the first term and the rule for generating subsequent terms.
Recursive Definition:
- Base Case: The first term of the sequence is defined as:
a(1) = 11 - Recursive Case: For all integers n > 1, each term can be defined based on the preceding term. In this case, we notice that each term decreases by 3, and then this decrement alternates slightly:
- a(n) = a(n-1) – 3, for n = 2 (which gives us 11 – 3 = 8)
- a(n) = a(n-1) – 3, for n = 3 (which gives us 8 – 3 = 5)
- a(n) = a(n-1) – 3, for n = 4 (which gives us 5 – 3 = 2)
Thus, to summarize, the recursive definition for the sequence is:
a(1) = 11
a(n) = a(n-1) - 3, for n > 1
This definition captures the structure of the sequence, allowing us to generate any term by knowing the previous one.