Summary
Highlights
Typically, sequences are defined using an nth term (e.g., u_n = 3n - 2). This 'position-to-term rule' allows you to find any term directly by substituting its position (n) into the formula. This method is efficient for finding any term in the sequence.
Not all sequences can be defined by an nth term. Some are defined by a 'term-to-term rule,' which is part of an inductive definition. An inductive definition requires two components: a first term (u_1) and a recurrence relation.
An inductive definition consists of a first term (e.g., u_1 = 3) and a recurrence relation. The recurrence relation is an instruction that tells you how to get the next term from the current term (e.g., u_n+1 = 2u_n - 3). Both parts are necessary to define the sequence.
Using the first term and the recurrence relation, you can generate the subsequent terms of the sequence. For example, if u_1 = 3 and u_n+1 = 2u_n - 3, then u_2 = 2(3) - 3 = 3, and u_3 = 2(3) - 3 = 3, leading to a constant sequence in this specific example.
Changing the recurrence relation slightly can drastically alter the sequence, demonstrating the flexibility of inductive definitions. However, a significant drawback is that to find a distant term, like the 100th term, you must calculate all preceding terms, unlike with an nth term formula.