Sequences

Numerical Sequence

In mathematics, a sequence is more than just a pattern; it's an ordered list of numbers where each number has a specific position (its place in the list). To work with sequences effectively, we use a special notation to distinguish a term's position from its value.

Definition Numerical Sequence

A numerical sequence is an ordered list of numbers, called terms. We use the notation $u_n$ to describe a term in the sequence.

The subscript $n$ is the index, which tells us the position of the term (often starting from $n=0$). The index $n$ usually takes integer values: $0,1,2,\dots$
$u_n$ represents the value of the term at that specific position.

So, $u_0$ is the value of the term at position $0$, $u_1$ is the value at position $1$, and so on.

Index ($n$)	0	1	2	$\dots$
Term ($u_n$)	$u_0$	$u_1$	$u_2$	$\dots$

Example

Given the sequence defined by the table below, find the value of the term $u_4$.

$n$	0	1	2	3	4	5	$\dots$
$u_n$	3	5	7	9	11	13	$\dots$

Answer

To find $u_4$, we look in the table for the column where the index is $n=4$. The value in the row below it is $11$.
Therefore, $u_4 = 11$.

Recursive Definition

Discover

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:

Here the sequence is indexed starting from $n=0$: $u_0 = 2$, $u_1 = 5$, $u_2 = 8$, $u_3 = 11$, etc.We can describe the relationship between the terms using formal notation:

$5 = 2\textcolor{colordef}{+3} \longrightarrow u_1 = u_0\textcolor{colordef}{+3} \longrightarrow u_{0+1} = u_0+3$
$8 = 5\textcolor{colordef}{+3} \longrightarrow u_2 = u_1\textcolor{colordef}{+3} \longrightarrow u_{1+1} = u_1+3$
$11 = 8\textcolor{colordef}{+3} \longrightarrow u_3 = u_2\textcolor{colordef}{+3} \longrightarrow u_{2+1} = u_2+3$

This pattern shows that any term $u_{n+1}$ can be found by adding $3$ to the previous term $u_n$. We can generalize this relationship as a rule (valid for all integers $n \ge 0$):$$u_{n+1} = u_n + 3$$

Definition Recursive Definition

A recursive definition of a sequence includes two parts:

The first term (or initial term), denoted for example by $u_0$ or $u_1$. This is the starting point.
The recursive rule (or recurrence relation), which is a formula that connects the next term, $u_{n+1}$, to the current term, $u_n$, for all appropriate integers $n$ (such as $n \ge 0$ or $n \ge 1$).

Once these two parts are known, every term in the sequence can be calculated step by step.

Example

A sequence is defined recursively by:

$u_1 = 5$ (The first term is $5$).
$u_{n+1} = u_n + 3$ (The rule is to add $3$ to the previous term, for all integers $n \ge 1$).

Find the first five terms of this sequence.

Answer

Let's build the sequence step-by-step using the recursive definition.

1$^{\text{st}}$ term: The starting term is given: $\boldsymbol{u_1 = 5}$.
2$^{\text{nd}}$ term: Use the rule with $n=1$. $\boldsymbol{u_2} = u_1 + 3 = 5 + 3 = 8$.
3$^{\text{rd}}$ term: Use the rule with $n=2$. $\boldsymbol{u_3} = u_2 + 3 = 8 + 3 = 11$.
4$^{\text{th}}$ term: Use the rule with $n=3$. $\boldsymbol{u_4} = u_3 + 3 = 11 + 3 = 14$.
5$^{\text{th}}$ term: Use the rule with $n=4$. $\boldsymbol{u_5} = u_4 + 3 = 14 + 3 = 17$.

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

Explicit Definition

While a recursive rule tells you how to get from one term to the next, it is not very efficient if you want to find a term far into the sequence (like the $100^{\text{th}}$ term), because you would have to calculate all the terms before it.
An alternative and often more powerful way to define a sequence is with an explicit rule. An explicit rule is a formula that gives the value of $u_n$ directly when you know the position $n$ in the sequence. You choose $n$, and the rule gives you $u_n$ immediately.

Definition Explicit Rule

An explicit rule (or explicit formula) defines the $n$th term of a sequence, $u_n$, directly as a function of its position $n$.$$u_n = f(n)$$

Note

The key advantage is that it allows us to calculate any term in the sequence directly, without first having to find all the previous terms one by one.

Example

Consider the sequence defined by the explicit formula:$$u_n = 3n + 2$$Calculate $u_{100}$.

Answer

To find the value of $u_{100}$, we substitute $n = 100$ into the explicit formula:$$\begin{aligned}[t]u_{100} &= 3(100) + 2 \\ &= 300 + 2 \\ &= \boldsymbol{302}\end{aligned}$$We did not need to know any of the previous terms in the sequence.

Arithmetic Sequence

An arithmetic sequence is the most common type of sequence that follows a recursive rule. It is defined by a starting term and a constant change between consecutive terms.

Discover

Let's consider an arithmetic sequence with a first term $u_1 = 5$ and a common difference $d = 3$.
We can write out the first few terms by repeatedly adding 3:

$u_1 = 5$
$u_2 = 5 + 3$
$u_3 = 5 + 3 + 3$
$u_4 = 5 + 3 + 3 + 3$

Let's rewrite this using multiplication to see the pattern:

$u_1 = 5 + \textcolor{colordef}{0} \times 3$
$u_2 = 5 + \textcolor{colordef}{1} \times 3$
$u_3 = 5 + \textcolor{colordef}{2} \times 3$
$u_4 = 5 + \textcolor{colordef}{3} \times 3$

The pattern is: to find the $n$th term ($u_n$), we start with $u_1$ and add the common difference $d$ exactly $\textcolor{colordef}{(n-1)}$ times. This gives us the explicit formula:$$u_n = u_1 + (n-1)d$$Now we can find $u_{10}$ directly:$$u_{10} = u_1 + (10-1)d = 5 + (9 \times 3) = 5 + 27 = 32$$This is much faster than calculating each term one by one!

Definition Arithmetic Sequence

An arithmetic sequence is a sequence where each term after the first is found by adding a constant value to the previous term. This constant is called the common difference, denoted by $d$.

Recursive Definition: The rule for finding the next term from the previous one is:$$u_{n+1} = u_n + d$$
Explicit Formula: We can also find any term directly using its position, $n$.
- If the sequence starts with $u_1$: $$u_n = u_1 + (n-1)d$$
- If the sequence starts with $u_0$: $$u_n = u_0 + nd$$

Example

An arithmetic sequence has a first term $u_1 = 5$ and a common difference $d = 3$.
Find the first five terms of this sequence.

Answer

The recursive rule is $u_{n+1} = u_n + 3$. We start with $u_1 = 5$ and apply the rule repeatedly.

$u_1 = \boldsymbol{5}$ (given)
$u_2 = u_1 + 3 = 5 + 3 = \boldsymbol{8}$
$u_3 = u_2 + 3 = 8 + 3 = \boldsymbol{11}$
$u_4 = u_3 + 3 = 11 + 3 = \boldsymbol{14}$
$u_5 = u_4 + 3 = 14 + 3 = \boldsymbol{17}$

The first five terms are: $5, 8, 11, 14, 17$.

Geometric Sequence

A geometric sequence is another fundamental type of sequence that follows a recursive rule. It is defined by a starting term and a constant multiplicative factor: to get from one term to the next, you always multiply by the same number.

Discover

Let's consider a geometric sequence with a first term $u_1 = 2$ and a common ratio $r = 3$.
We can write out the first few terms by repeatedly multiplying by 3:

$u_1 = 2$
$u_2 = 2 \times 3$
$u_3 = 2 \times 3 \times 3$
$u_4 = 2 \times 3 \times 3 \times 3$

Let's rewrite this using exponents to see the pattern:

$u_1 = 2 \times 3^{\textcolor{colordef}{0}}$
$u_2 = 2 \times 3^{\textcolor{colordef}{1}}$
$u_3 = 2 \times 3^{\textcolor{colordef}{2}}$
$u_4 = 2 \times 3^{\textcolor{colordef}{3}}$

The pattern is: to find the $n$th term ($u_n$), we start with $u_1$ and multiply by the common ratio $r$ exactly $\textcolor{colordef}{(n-1)}$ times. This gives us the explicit formula:$$u_n = u_1 \times r^{n-1}$$Now we can find $u_{10}$ directly:$$u_{10} = u_1 \times r^{10-1} = 2 \times 3^9 = 2 \times 19683 = 39366$$This is much faster than calculating each term one by one!

Definition Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant non-zero value. This constant is called the common ratio, denoted by $r$.

Recursive Definition: The rule for finding the next term from the previous one is:$$u_{n+1} = r \times u_n $$
Explicit Formula: We can also find any term directly using its position, $n$.
- If the sequence starts with $u_1$: $$u_n = u_1 \times r^{n-1}$$
- If the sequence starts with $u_0$: $$u_n = u_0 \times r^n$$

Example

A geometric sequence has a first term $u_1 = 2$ and a common ratio $r = 2$.
Find the first five terms of this sequence.

Answer

The recursive rule is $u_{n+1} = u_n \times 2$. We start with $u_1 = 2$ and apply the rule repeatedly.

$u_1 = \boldsymbol{2}$ (given)
$u_2 = 2 \times u_1 = 2 \times 2 = \boldsymbol{4}$
$u_3 = 2 \times u_2 = 2 \times 4 = \boldsymbol{8}$
$u_4 = 2 \times u_3 = 2 \times 8 = \boldsymbol{16}$
$u_5 = 2 \times u_4 = 2 \times 16 = \boldsymbol{32}$

The first five terms are: $2, 4, 8, 16, 32$.

Series

While a sequence is a list of numbers, a series is what you get when you add those numbers together. Every sequence has a corresponding series. We are often interested in the partial sum of a sequence, which is the sum of a specific number of its terms, starting from the beginning.

Definition Series and Partial Sum

A series is the sum of the terms in a sequence. The partial sum, denoted $\boldsymbol{S_n}$, is the sum of the terms of a sequence up to a specific index $n$.

If a sequence starts at $u_0$, the partial sum $S_n$ is the sum of the first $\textcolor{colordef}{(n+1)}$ terms: $$S_n = u_0 + u_1 + u_2 + \ldots + u_n = \sum_{i=0}^n u_i$$
If a sequence starts at $u_1$, the partial sum $S_n$ is the sum of the first $\textcolor{colordef}{n}$ terms: $$S_n = u_1 + u_2 + u_3 + \ldots + u_n = \sum_{i=1}^n u_i$$

Sum of an Arithmetic Sequence

Discover

How can we efficiently calculate the sum of the first 20 terms of an arithmetic sequence starting with $u_0 = 5$ and a common difference of $d=5$?$$S_{19} = \overbrace{5 + 10 + 15 + \ldots + 90 + 95 + 100}^{\text{20 terms}}$$The trick, famously discovered by Gauss as a child, is to write the sum twice, once in forward order and once in reverse order, and then add the two lines together term by term.$$\begin{array}{l r c c c c c c c}S_{19} & = & 5 & + & 10 & + & \ldots & + & 100 \\ S_{19} & = & 100 & + & 95 & + & \ldots & + & 5 \\ \hline2S_{19} & = & 105 & + & 105 & + & \ldots & + & 105 \\ \end{array}$$Each pair of terms adds up to the same value ($5+100=105$, $10+95=105$, etc.). There are 20 identical sums in $2S_{19}$.$$2S_{19} = 20 \times (5+100)$$Therefore, the sum is:$$S_{19} = 20 \times\frac{5+100}{2} = 1050$$This method reveals the general formula:$$S_n = \frac{\text{Number of terms}}{2} \times (\text{First term} + \text{Last term})$$

Proposition Sum of an Arithmetic Sequence

The sum of the first terms of an arithmetic sequence is given by the formula:$$S_n = \text{Number of terms} \times \frac{(\text{First term} + \text{Last term})}{2}$$

If a sequence starts at $u_0$ and ends at $u_n$, there are $\boldsymbol{n+1}$ terms. The formula is:$$S_n = \frac{n+1}{2}(u_0 + u_n)$$
If a sequence starts at $u_1$ and ends at $u_n$, there are $\boldsymbol{n}$ terms. The formula is:$$S_n = \frac{n}{2}(u_1 + u_n)$$

Sum of a Geometric Sequence

Discover

How can we efficiently calculate the sum of the first 10 terms of a geometric sequence starting with $u_0 = 3$ and a common ratio of $r=2$?$$S_9 = 3 + 6 + 12 + 24 + \ldots + 768 + 1536$$The trick is to multiply the entire sum by the common ratio, $r=2$, and then subtract the original sum from this new one.
Let's write the original sum:$$S_9 = 3 + 6 + 12 + \ldots + 768 + 1536$$Now, let's multiply every term by the common ratio, 2:$$2S_9 = 6 + 12 + 24 + \ldots + 1536 + 3072$$Notice that most of the terms in $S_9$ and $2S_9$ are identical. If we subtract the first equation from the second, these terms will cancel out:$$\begin{array}{lrccccccccc} 2S_9 & = & & & 6 & + & 12 & + \ldots + & 1536 & + & 3072 \\ -\, S_9 & = & -(3 & + & 6 & +&12 & + \ldots + & 1536) \\ \hline(2-1)S_9 & = & -3 & + & 0 & + &0&+\ldots + & 0 & + & 3072 \\ \end{array}$$This method reveals the general formula for the sum of any geometric sequence.$$\begin{aligned}S_n - rS_n &= u_0 - u_0 \times r^{n+1}\\ S_n(1-r) &= u_0(1-r^{n+1})\\ S_n &= u_0 \left( \frac{1 - r^{n+1}}{1 - r}\right)\\ \end{aligned}$$

Proposition Sum of a Geometric Sequence

The sum of the first terms of a geometric sequence is given by the formula:$$S_n = \text{First term} \times \frac{1 - (\text{common ratio})^{\text{Number of terms}}}{1 - \text{common ratio}}$$

If a sequence starts at $u_0$ and has a common ratio $r$, the sum of the first $\boldsymbol{n+1}$ terms is:$$S_n = u_0 \left( \frac{1 - r^{n+1}}{1 - r} \right)$$
If a sequence starts at $u_1$ and has a common ratio $r$, the sum of the first $\boldsymbol{n}$ terms is:$$S_n = u_1 \left( \frac{1 - r^{n}}{1 - r} \right)$$

These formulas are valid for any common ratio $r \neq 1$.

Index (\(n\))	0	1	2	\(\dots\)
Term (\(u_n\))	\(u_0\)	\(u_1\)	\(u_2\)	\(\dots\)

\(n\)	0	1	2	3	4	5	\(\dots\)
\(u_n\)	3	5	7	9	11	13	\(\dots\)