Mathematical Induction

Inductive Definition of Sequences

1. In general, a sequence is defined inductively by:
1. The value of the first term ,
2. A recurrence relation between two consecutive terms and (for ).
This way of defining a sequence by specifying the first few terms and the relationships between consecutive terms is called the inductive definition of a sequence.
2. Inductive Definition of Arithmetic and Geometric Sequences
1. Inductive definition of an arithmetic sequence:
   A sequence defined inductively as is an arithmetic sequence with the first term and common difference .
2. Inductive definition of a geometric sequence:
   A sequence defined inductively as is a geometric sequence with the first term and common ratio .

Mathematical Induction

To prove that a proposition holds for all natural numbers , the following two steps need to be shown:

1. The proposition holds for
2. Assuming that the proposition holds for , it must also hold for

This method of proving is called mathematical induction.