Arithmetic Sequences

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.

\( \begin{align} \displaystyle
u_{10} &= u_{1} + 9d = 100 \cdots (1) \\
u_{15} &= u_{1} + 14d = 175 \cdots (2) \\
14d – 9d &= 175 – 100 &(2)-(1) \\
5d &= 75 \\
d &= 15 \\
u_{1} + 9 \times 15 &= 100 &\text{substitute } d=25 \text{ into } (1) \\
u_{1} + 135 &= 100 \\
u_{1} &= -35 \\
u_{n} &= -35 + (n-1) \times 15 \\
&= -35 + 15n – 15 \\
\therefore u_{n} &= 15n-50 \\
\end{align} \)

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