We will now explore recursively defined sequences of complex numbers. Recursively defined sequences are sequences in which subsequent terms are constructed based on preceding terms using a specific set of rules or formulas, allowing each term to be defined as a function of its predecessors.
Recursive Sequence
A recursive relationship is a formula which relates the next value, [latex]{{z}_{n+1}}[/latex], in a sequence to the previous value, [latex]{{z}_{n}}[/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[/latex].
The sequence of values produced is the recursive sequence.
In mathematics, we often use subscripts to denote specific elements or terms in a sequence or set. Imagine you have a list of data entries represented by the symbol [latex]x[/latex]. These entries can be labeled as [latex]x_1,x_2,x_3, ...[/latex], where the subscript indicates the position of the entry in the list.
For instance, [latex]x_1[/latex] refers to the first entry, and [latex]x_2[/latex] refers to the second entry, and so on. We sometimes start our list with a “zeroth” term, denoted as [latex]x_0[/latex]. When we want to talk about a general term in the list, we use [latex]x_n[/latex], where [latex]n[/latex] represents any position in the list. This notation also allows us to refer to the terms before and after [latex]x_n[/latex] as [latex]x_{n-1}[/latex] and [latex]x_{n+1}[/latex], respectively.
How To: Apply a Recursive Sequence
Identify the Initial Term: Start by determining the first term of the sequence, often denoted as [latex]a_0[/latex] or [latex]a_1[/latex], provided in the sequence definition. This term serves as the starting point for building the rest of the sequence.
Understand the Recursive Formula: Look at the recursive relationship that defines how to find each term from the previous term(s). The formula will generally be given in the form of [latex]a_{n+1}=f(a_n)[/latex], where [latex]f[/latex] represents a function.
Apply the Formula: Use the formula to calculate the next term in the sequence by substituting the previous term into the formula.
Repeat the Process: Continue applying the recursive formula to each new term to find subsequent terms as needed.
Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}+2, {{z}_{0}}=4[/latex], generate several terms of the recursive sequence.
We are given the starting value, [latex]{{z}_{0}}=4[/latex]. The recursive formula holds for any value of [latex]n[/latex], so if [latex]n = 0[/latex], then [latex]{{z}_{n+1}}={{z}_{n}}+2[/latex] would tell us [latex]{{z}_{0+1}}={{z}_{0}}+2[/latex], or more simply, [latex]{{z}_{1}}={{z}_{0}}+2[/latex]. Notice this defines [latex]{{z}_{1}}[/latex] in terms of the known [latex]{{z}_{0}}[/latex], so we can compute the value:
[latex]{{z}_{1}}={{z}_{0}}+2=4+2=6[/latex].
Now letting [latex]n = 1[/latex], the formula tells us [latex]{{z}_{1+1}}={{z}_{1}}+2[/latex], or [latex]{{z}_{2}}={{z}_{1}}+2[/latex]. Again, the formula gives the next value in the sequence in terms of the previous value.
[latex]{{z}_{2}}={{z}_{1}}+2=6+2=8[/latex]
Continuing,
[latex]{{z}_{3}}={{z}_{2}}+2=8+2=10[/latex]
[latex]{{z}_{4}}={{z}_{3}}+2=10+2=12[/latex]
The previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.
Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}\cdot{i}+(1-i), {{z}_{0}}=4[/latex], generate several terms of the recursive sequence.
We are given [latex]{{z}_{0}}=4[/latex]. Using the recursive formula: