Example 5 "Tossing a coin continuously until two consecutive heads appear" occurs in all possible ways on the $n(n \geqslant 2)$-th toss.

How many such ways are there?
Explanation: Let the number of methods that satisfy the conditions be $a_{n}$.
When $n=2$, there is only 1 case of "positive positive", $a_{2}=1$;

When $n=3$, there is only 1 case of "negative positive positive", $a_{3}=1$.

For $n \geqslant 4$, the number of methods $a_{n}$ where "positive positive" appears exactly at the $n$-th toss includes two scenarios:
(1) The 1st toss is negative, and "positive positive" appears in the subsequent $n-1$ tosses, with $a_{n-1}$ methods;
(2) The 1st toss is positive, then the 2nd toss must be negative, and "positive positive" appears in the subsequent $n-2$ tosses, with $a_{n-2}$ methods.
Thus, we have $\left\{\begin{array}{l}a_{n}=a_{n-1}+a_{n-2}, \\ a_{2}=1, a_{3}=1 .\end{array}\right.$
Therefore, the problem of finding the number of methods is transformed into finding the general term of the Fibonacci sequence, also using the recursive method.
1.2 Understanding of the Recursive Method

From the above example, we can make an initial definition of recursive relations, the recursive method, and how to use the recursive method.
1.2.1 Recursive Relation

Let $f(1), f(2), \cdots, f(n), \cdots$ represent the various states of a changing process. If state $f(n)$ is dependent on previous states $f(n-1), f(n-2)$, etc., and can be expressed as an equation or inequality (see Example 7)
$f(n)=F(f(n-1), f(n-2), \cdots, f(1))$

or $f(n) \geqslant F(f(n-1), f(n-2), \cdots, f(1))$
or $f(n) \leqslant F(f(n-1), f(n-2), \cdots, f(1))$.
Then such a relationship is called a recursive relation, and when it is an equality, it is commonly known as a recursive equation or recursive formula.

In this case, one can start from some initial conditions and gradually deduce the result at any point in time. That is, the initial conditions and the recursive relation (both are indispensable) can completely determine a process (for example, in Example 1, from $a_{1}=1, a_{2}=1$ and the recursive relation (1), the Fibonacci sequence can be written as: $1,1,2,3,5,8,13,21,34,55,89,144,233$, $377, \cdots$. The answer is $\left.a_{13}=233\right)$.