1, accumulation method: For the recurrence relation in the form of an-an- 1=d (constant), we can get the general term formula by accumulation. For example, for the series 1, 3,6, 10, 15 ..., we can see that each number is the sum of the previous number and 1, that is, an-an- 1= 1. By accumulation, we can get an=n(n+ 1)/2.

2. Iteration method: For the recurrence relation in the form of an=an- 1+f(n), we can get the general term formula through continuous iteration. For example, for the series 1, 3,7,13,21..., we can see that each number is the sum of the previous number and 2, that is, an=an- 1+2. Through continuous iteration, an=(n- 1)*2+ 1 can be obtained.

3. Construction method: For the recurrence relation that does not meet the above two methods, we can use the construction method to obtain the general term formula. For example, for the recurrence relation in the form of an=an- 1+n, we can obtain the general term formula by constructing it. First, we construct a new sequence bn=an-n, and we can find that this sequence bn is arithmetic progression. Then the general term formula of bn is obtained by using arithmetic progression's general term formula. Finally, the general term formula of an can be obtained by adding n.

The application of finding the general term from recursive relation in mathematics;

1, Fibonacci sequence: Fibonacci sequence is a classical recursive relational sequence, which is defined as follows: a( 1)= 1, a(2)= 1, a (n) = a (n-1. 2。 This series is obtained by iteration according to the recursive relation, and the value of each item can be easily obtained through the recursive relation. Fibonacci sequence has important applications in mathematics, such as solving some optimization problems, graph theory and other fields.

2. Difference equation: In physics, engineering and other fields, the problem of difference equation is often encountered. The difference equation is a recursive relation, and the state of each moment can be obtained conveniently through the recursive relation. For example, in electromagnetic field calculation, fluid mechanics and other fields, the problem of difference equation is often encountered, and the state of the object at different times can be obtained by solving the difference equation.

3. Cryptography: In cryptography, recursive relations are often used to generate passwords. For example, Caesar's cipher is a simple shift cipher, which moves each letter in the text backwards by a certain position to get the ciphertext. The position of this shift is obtained by recursive relation. In addition, in more complex cryptography, such as RSA encryption algorithm, recursive relations are also used to generate keys.