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.