## Algebraic Definition

An $\textit{Arithmetic Sequence}$ is a sequence in which each term differs from the previous one by the same fixed number, which is often called $\textit{common difference}$. It can also be referred to as an $\textit{arithmetic progression}$.

A sequence in mathematics is an ordered set of numbers.

An $\textit{arithmetic sequence}$ is one in which:- the difference between any two successive terms is the same
- the next term in the sequence is found by adding the same number

Arithmetic Sequences Formula

For example:- $1, 3, 5, 7, 9, \cdots$

the common difference is $2$ - $35, 30, 27, 24, \cdots$

the common difference is $-3$

$\{u_{n}\}$ is $\textit{arithmetic}$ if and only if $u_{n+1} - u_{n} = d$ for all positive $n$ where $d$ is a the common difference.- $3-1=2$
- $5-3=2$
- $7-5=2$
- $u_{n+1} - u_{n} =2$

There the sequence $1, 3, 5, 7, 9, \cdots$ is arithmetic.## Arithmetic Mean

If $a$, $b$ and $c$ are any consecutive terms of an arithmetic sequence then:
\( \begin{align} \displaystyle
u_{2} - u_{1} &= u_{3} - u_{2} &\text{equating common difference} \\
b-a &= c-b \\
2b &= a+c \\
\therefore b &= \dfrac{a+c}{2} \\
\end{align} \)

This means that the middle term is the $\textit{arithmetic mean}$ of the terms on either side of it.## General Term Formula

Suppose that the first term of an arithmetic sequence is $u_{1}$, or $a$ and the common difference is $d$.
\( \begin{align} \displaystyle
u_{2} &= u_{1} + d \\
&= u_{1} + (2-1)d \\
u_{3} &= u_{2} + d \\
&= (u_{1} + d) + d \\
&= u_{1} + 2d \\
&= u_{1} + (3-1)d \\
u_{4} &= u_{3} + d \\
&= (u_{1} + 2d) +d \\
&= u_{1} + 3d \\
&= u_{1} + (4-1)d \\
&\cdots \\
\therefore u_{n} &= u_{1} + (n-1)d \\
\text{or}\\
\therefore T_{n} &= a + (n-1)d \\
\end{align} \)

If we are given only two terms of an arithmetic sequence, we are able to use the rule $u_{n}=u_{1}+(n-1)d$ to set up two simultaneous equations to find the value of $u_{1}$, or $a$ and $d$ and hence write down the rule for the arithmetic sequence.### Example 1

Show that the sequence $3, 10, 17, 24, 31, \cdots$ is arithmetic. \( \begin{align} \displaystyle
10 - 3 &= 7 \\
17 - 10 &= 7 \\
24 - 17 &= 7 \\
31 - 24 &= 7 \\
\end{align} \)

The difference between successive terms is constant.

So the sequence is arithmetic with $u_{1}=3$ and $d=7$.

### Example 2

Find a formula for the general term of $5, 8, 11, 14, 17, \cdots$ \( \begin{align} \displaystyle
17 - 14 &= 3 \\
14 - 11 &= 3 \\
11 - 8 &= 3 \\
8 - 5 &= 3 \\
\end{align} \)

The difference between successive terms is constant.

So the sequence is arithmetic with $u_{1}=5$ and $d=3$.

\( \begin{align} \displaystyle
u_{n} &= u_{1} + (n-1) \times d \\
u_{n} &= 5 + (n-1) \times 3 \\
&= 5 + 3n - 3 \\
\therefore u_{n} &= 3n + 2 \\
\end{align} \)

### Example 3

Find the 150^{th} term of the sequence: $2, 6, 10, 14, \cdots$. \( \begin{align} \displaystyle
14 - 10 &= 4 \\
10 - 6 &= 4 \\
6 - 2 &= 4 \\
\end{align} \)

The difference between successive terms is constant.

So the sequence is arithmetic with $u_{1}=2$ and $d=4$.

\( \begin{align} \displaystyle
u_{n} &= u_{1} + (n-1) \times d \\
u_{n} &= 2 + (n-1) \times 4 \\
&= 2 + 4n - 4 \\
u_{n} &= 4n - 2 \\
u_{150} &= 4 \times 150 - 2 \\
\therefore u_{150} &= 598 \\
\end{align} \)

### Example 4

Is $83$ a term of the sequence: $4, 7, 10, 13, \cdots$? \( \begin{align} \displaystyle
13 - 10 &= 3 \\
10 - 7 &= 3 \\
7 - 4 &= 3 \\
\end{align} \)

The difference between successive terms is constant.

So the sequence is arithmetic with $u_{1}=4$ and $d=3$.

\( \begin{align} \displaystyle
u_{n} &= u_{1} + (n-1) \times d \\
u_{n} &= 4 + (n-1) \times 3 \\
&= 4 + 3n - 3 \\
u_{n} &= 3n + 1 \\
3n + 1 &= 83 \\
3n &= 82 \\
n &= 82 \div 3 \\
&= 27.333 \cdots \\
\end{align} \)

$n$ must be a positive integer, thus $83$ is $\textit{not}$ a term of the sequence.

### Example 5

Which term is $191$ of the sequence $11, 14, 17, 20, \cdots$? \( \begin{align} \displaystyle
20 - 17 &= 3 \\
17 - 14 &= 3 \\
14 - 11 &= 3 \\
\end{align} \)

The difference between successive terms is constant.

So the sequence is arithmetic with $u_{1}=11$ and $d=3$.

\( \begin{align} \displaystyle
u_{n} &= u_{1} + (n-1) \times d \\
u_{n} &= 11 + (n-1) \times 3 \\
&= 11 + 3n - 3 \\
u_{n} &= 3n + 8 \\
3n + 8 &= 191 \\
3n &= 183 \\
n &= 183 \div 3 \\
&= 61 \\
\end{align} \)

Therefore $191$ is $61$^{st} term of the sequence.

### Example 6

If $u_{10}=100$ and $u_{15}=175$, find the $n$^{th} term for the arithmetic sequence. \( \begin{align} \displaystyle
u_{10} &= u_{1} + 9d = 100 \cdots (1) \\
u_{15} &= u_{1} + 14d = 175 \cdots (2) \\
14d - 9d &= 175 - 100 &(2)-(1) \\
5d &= 75 \\
d &= 15 \\
u_{1} + 9 \times 15 &= 100 &\text{substitute } d=25 \text{ into } (1) \\
u_{1} + 135 &= 100 \\
u_{1} &= -35 \\
u_{n} &= -35 + (n-1) \times 15 \\
&= -35 + 15n - 15 \\
\therefore u_{n} &= 15n-50 \\
\end{align} \)