Final answer:
The recursion rule for the sequence 1, 3, 6, 10, 15, ... is T(n) = T(n - 1) + n, with T(1) = 1, which defines the sequence of triangular numbers and determines each term based on the sum of the previous term and the position of the term.
Step-by-step explanation:
The sequence provided, 1, 3, 6, 10, 15, ..., is known as a triangular number sequence.
Each number represents the sum of the natural numbers up to a certain point.
The recursive rule for this sequence can be defined by the nth term relation, which is
T(n) = T(n - 1) + n
where T(n) is the nth term of the sequence and T(n - 1) is the previous term.
To initialize the recursion, we specify that T(1) = 1, because the first term of the sequence is 1.
As an example, let's calculate T(5), which is the 5th term in the sequence.
We know T(4) = 10, so T(5) = T(4) + 5 = 10 + 5 = 15, which corresponds to the given sequence.