Recurrence Relations
Neil Trivedi
Teacher
A recurrence relation (also known as an inductive relation) tells us how to find a term in a sequence from the previous term(s). The definition must also include the value of a term of the sequence. We can work out the values of other terms using the given term. Usually, we are given the first term which allows us to find the second term. Then, the third term can be worked out from the second term, and so on.
An arithmetic sequence follows the form:
where we add a constant to the previous term to get the next term.
A geometric sequence follows the form:
where we multiply the previous term by a constant to get the next term.
Example 1:
A sequence is defined by the recurrence relation:
a) Write down the first four terms of the sequence.
Single Step: Substitute each previous term into the recurrence formula.
Thus, the first four terms of the sequence are and .
b) Find .
Single Step: Sum the first terms of the sequence.
No answer provided.
Example 2:
A sequence is defined by the recurrence relation:
a) Write down the first three terms of the sequence.
Single Step: This is a geometric sequence as we are working out of the previous term. Substitute each previous term into the recurrence formula.
Thus, the first three terms of the sequence are and .
b) Find .
Single Step: Use sum to infinity formula for geometric series, where and .
No answer provided.
Increasing, Decreasing and Periodic Sequences
In an increasing sequence, each term is greater than the one before.
In a decreasing sequence, each term is smaller than the one before.
A periodic sequence repeats itself at regular intervals. The number of terms before the sequence repeats is called the period (also known as the order). For example, the sequence is periodic with a period of , since every terms, the sequence repeats.
Example 3:
For each sequence, state whether it is increasing, decreasing, or periodic. If the sequence is periodic, write down its order.
a)
This is an arithmetic sequence, where each term is obtained by adding to the previous term:
Since each term is greater than the previous one, this is an increasing sequence.
b)
Each term is obtained by squaring the previous term:
Since each term is smaller than the previous one, this is a decreasing sequence.
c)
This is a deductive formula, not a recurrence relation, since each term depends directly on .
The sequence will be as follows:
Since the sequence repeats every terms, it is a periodic sequence of order .
No answer provided.
Challenging Question