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

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.

Example 2

Find a formula for the general term of $5, 8, 11, 14, 17, \cdots$

Example 3

Find the 150th term of the sequence: $2, 6, 10, 14, \cdots$.

Example 4

Is $83$ a term of the sequence: $4, 7, 10, 13, \cdots$?

Example 5

Which term is $191$ of the sequence $11, 14, 17, 20, \cdots$?

Example 6

If $u_{10}=100$ and $u_{15}=175$, find the $n$th term for the arithmetic sequence.