How are recursive and explicit rules the same?

Recursive and explicit rules are different types of rules used to define sequences. They are not the same.

Recursive rules define a term in a sequence based on the previous term or terms. For example, the recursive rule for the Fibonacci sequence is $$F(n) = F(n-1) + F(n-2),$$ where \(F(1) = 1\) and \(F(2) = 1\).

Explicit rules define a term in a sequence using a formula that involves the position of the term in the sequence. For example, the explicit rule for the arithmetic sequence \(3, 7, 11, 15, 19\dots\) is given by:

$$a_n = 4n – 1$$.