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Sequences

Numerical Sequence

Definition Numerical Sequence
A numerical sequence, \((u_n)\) is an ordered list of numbers \((u_0,\,u_1,\,u_2,\dots)\) defined by a rule.
The number \(u_n\) is called the \(n\)th term of the sequence.
Example
What is \(u_4\) of this sequence?
\(n\) 0 1 2 3 4 5 \(\dots\)
\(u_n\) 3 5 7 9 11 13 \(\dots\)

\(u_4 = 11\).

Definition Using a Recursive Rule


Let’s consider a sequence where the first term is \(2\), and each term is obtained by adding \(3\) to the previous term. The terms are:
We observe:
  • \(5 = 2\textcolor{colordef}{+3} \longrightarrow u_1 = u_0\textcolor{colordef}{+3} \longrightarrow u_{0+1} = u_0+3\)
  • \(8 = 5\textcolor{colordef}{+3} \longrightarrow u_2 = u_1\textcolor{colordef}{+3} \longrightarrow u_{1+1} = u_1+3\)
  • \(11 = 8\textcolor{colordef}{+3} \longrightarrow u_3 = u_2\textcolor{colordef}{+3} \longrightarrow u_{2+1} = u_2+3\)
  • \(\vdots\)
  • So the rule is \(u_{n+1} = u_n + 3\)

Definition Recursive Rule
A sequence can be defined by:
  • the first term (starting number): \(u_0\)
  • a recursive rule that tells how to obtain each term from the previous one:
    \(u_{n+1}=\) expression in \(u_n\)
Example
Write the recursive rule when each term is obtained by adding \(2\) to the previous term.

$$u_{n+1}=u_n+2$$

Definition Using an Explicit Rule

Definition Explicit Rule
A sequence can also be defined by an explicit rule (or explicit formula), which gives a direct formula for the \(n\)th term in terms of \(n\):$$u_n = \text{expression in } n$$
Example
Consider the sequence defined by the explicit formula: \(u_n = 3n + 2\).
Write the first five terms of this sequence.

  • For \(n=0\): $$ \begin{aligned}[t] u_0 &= 3 \times 0 + 2 \\ &= 0 + 2 \\ &= 2 \end{aligned} $$
  • For \(n=1\): $$ \begin{aligned}[t] u_1 &= 3 \times 1 + 2 \\ &= 3 + 2 \\ &= 5 \end{aligned} $$
  • For \(n=2\): $$ \begin{aligned}[t] u_2 &= 3 \times 2 + 2 \\ &= 6 + 2 \\ &= 8 \end{aligned} $$
  • For \(n=3\): $$ \begin{aligned}[t] u_3 &= 3 \times 3 + 2 \\ &= 9 + 2 \\ &= 11 \end{aligned} $$
  • For \(n=4\): $$ \begin{aligned}[t] u_4 &= 3 \times 4 + 2 \\ &= 12 + 2 \\ &= 14 \end{aligned} $$
So the first five terms are: \(2,\ 5,\ 8,\ 11,\ 14\).

Arithmetic Sequences

Definition Arithmetic Sequence
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant is called the common difference and is denoted by \(d\).
  • The recursive rule is: \(u_{n+1} = u_n + d\)
  • The explicit formula is: \(u_n = u_0 + n d\)
Example
Determine if the sequence \((2, 5, 8, 11, 14, \dots)\) is arithmetic and find the common difference \(d\) if it is.

The differences between consecutive terms are:\(5 - 2 = 3\), \(8 - 5 = 3\), \(11 - 8 = 3\), \(14 - 11 = 3\).
Since the difference is constant and equal to \(3\), the sequence is arithmetic with \(d = 3\).

Geometric Sequences

Definition Geometric Sequence
An geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant is called the common ratio and is denoted by \(r\).
  • The recursive rule is: \(u_{n+1} = u_n \times r\)
  • The explicit formula is: \(u_n = u_0 \times r^n\)
Example
Determine if the sequence \((2, 6, 18, 54, 162, \dots)\) is geometric and find the common ratio \(r\) if it is.

The ratios between consecutive terms are:\(6 \div 2 = 3\), \(18 \div 6 = 3\), \(54 \div 18 = 3\), \(162 \div 54 = 3\).
Since the ratio is constant and equal to \(3\), the sequence is geometric with \(r = 3\).

Series

Definition Series
A series is the sum of the terms of a sequence.$$\begin{aligned}S_n &= u_0 + u_1 + u_2 + \ldots + u_n\\ &=\sum_{i=0}^n u_i \\ \end{aligned}$$

Sum of an Arithmetic Sequence


We want to calculate$$S_{19} = \overbrace{5 + 10 + 15 + \ldots + 90 + 95 + 100}^{20~\text{terms}}$$The first term is \(u_0 = 5\) and the last term is \(u_{19} = 100\). However, we can also write:$$\begin{aligned}&&S_{19}&= & 5 &+& 10 &+& 15 &+& \ldots &+& 90 &+& 95 &+& 100 \\ +\,\,\,&&S_{19}&= & 100 &+& 95 &+& 90 &+& \ldots &+& 15 &+& 10 &+& 5 && \text{(reversing the terms)}\\ \hline&&2S_{19}&= & 105 &+& 105 &+& 105 &+& \ldots &+& 105 &+& 105 &+& 105 && \text{(adding)}\\ &&2S_{19}&= & 20 \times 105 &&&&&&&&&&&&&& \text{(20 times the same term)}\\ \end{aligned}$$So$$S_{19} = \dfrac{20}{2} \times 105 = 1050$$

Proposition Sum of an Arithmetic Sequence
The sum of an arithmetic sequence is$$S_n = \frac{n+1}{2}\left(u_0 + u_n\right)$$

Sum of an Geometric Sequence

Proposition Sum of a Geometric Sequence
The sum of a geometric sequence is$$S_n = u_0 \cdot \frac{1 - r^{n+1}}{1 - r}$$where \(r\) is the common ratio.