In mathematical logic and information technology, a recursive meaning (or inductive definition) is used to determine an item with regards to itself.
A recursive meaning of a operate describes principles of the features for some information with regards to the of the same operate for other information. For example, the factorial operate n! is determined by the rules
0! = 1.
(n+1)! = (n+1)n!.
This meaning is legitimate for all n, because the recursion gradually gets to the platform situation of 0. The meaning may also be believed of as providing a process explaining how to create the operate n!, beginning from n = 0 and continuing forward with n = 1, n = 2, n = 3 etc..
That such a meaning indeed describes a operate can be shown by introduction.

Definition: Arithmetic sequence or progression is a sequence of numbers such that the difference between two successive numbers is a constant. Geometric sequence or progression is a sequence of numbers such that the ratio between two successive numbers is a constant. Recursive formula is used to find the next term of the sequence using one or more preceding terms of the sequence

Recursive Formula Definition - Example Problems

See these solved problems - Recursive formula definition

Example 1: First term a1 = 39, common difference d = 15, what is the recursive formula of a arithmetic sequence?

Solution:

First term a1 = 39, common difference d = 15.

Second term a2 = 39 + 15 = 54

Third term a3 = 54 + 15 = 69

nth term an = an-1 + 15

Therefore, recursive formula of a arithmetic sequence is an = an-1 + 15 where a1 = 39.

Example 2: First term a1 = 58, common difference d = 21, what is recursive formula of a arithmetic sequence?

Solution:

First term a1 = 58, common difference d = 21.

Second term a2 = 58 + 21 = 79

Third term a3 = 79 + 21 = 100

nth term an = an-1 + 21

Therefore, recursive formula of a arithmetic sequence is an = an-1 + 21 where a1 = 58.

Example 3: First term a1 = 5, common ratio r = 11, what is the recursive formula of a geometric sequence?

Solution:

First term a1 = 5, common ratio r = 11.

Second term a2 = 5 * 11 = 55

Third term a3 = 55 * 11 = 605

nth term an = an-1 * 11

Therefore, recursive formula of a geometric sequence is an = an-1 * 11 where a1 = 5.

Example 4: First term a1 = 10, common ratio r = 20, what is the recursive formula of a geometric sequence?

Solution:

First term a1 = 10, common ratio r = 20.

Second term a2 = 10 * 20 = 200

Third term a3 = 200 * 20 = 4000

nth term an = an-1 * 20

Therefore, recursive formula of the sequence is an = an-1 * 20 where a1 = 10.

Recursive Formula Definition - Practice Problems

Problem 1: First term a1 = 35, common difference d = 25, what is the recursive formula of a arithmetic sequence?

Problem 2: First term a1 = 42, common ratio r = 18, what is the recursive formula of the geometric sequence?

Answer: 1) an = an-1 + 25, a1 = 35 2) an = an-1 * 18, a1 = 42