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Sequences

Numerical Sequence

Definition Numerical Sequence
A numerical sequence is a list of numbers.
  • The first number is called the 1\(^{\text{st}}\) term.
  • The second number is called the 2\(^{\text{nd}}\) term.
  • The third number is called the 3\(^{\text{rd}}\) term.
  • And so on.
Example
What is the 6\(^{\text{th}}\) term of this sequence?
\(n\) 1 2 3 4 5 6
\(n^{\text{th}}\) term 3 5 7 9 11 13

The 6\(^{\text{th}}\) term is \(13\).

Recursive Definition

Definition Recursive Definition
We can define a sequence recursively if we know:
  • the first term of the sequence,
  • the recursive rule that tells how to get from one term to the next.
Example
Find the first five terms in the sequence: start at 5 and add 3 each time.

$$5 \textcolor{olive}{\xrightarrow{\;+3\;}} 8 \textcolor{olive}{\xrightarrow{\;+3\;}} 11 \textcolor{olive}{\xrightarrow{\;+3\;}} 14 \textcolor{olive}{\xrightarrow{\,+3\,}} 17$$
The first five terms are: 5, 8, 11, 14, 17.

Arithmetic Sequence

Definition Arithmetic Sequence
An arithmetic sequence is a list of numbers in which the same value is added or subtracted each time to get the next term.
Example
What is the 6\(^{\text{th}}\) term of this sequence?
\(n\) 1 2 3 4 5 6
\(n^{\text{th}}\) term 3 5 7 9 11 ?

The 6\(^{\text{th}}\) term is \(13\), because each term increases by \(2\).

Geometric Sequence

Definition Geometric Sequence
A geometric sequence is a list of numbers in which the same value is multiplied or divided each time to get the next term.
Example
What is the 5\(^{\text{th}}\) term of this sequence?
\(n\) 1 2 3 4 5
\(n^{\text{th}}\) term 2 4 8 16 ?

The 5\(^{\text{th}}\) term is \(32\), because each term is multiplied by \(2\).