# Arithmetic Sequence | Maths Help

An Arithmetic Sequence is a sequence in which each term differs from the previous one by the same fixed number. It cal also be referred to as an arithmetic progression. For example,

• 2, 5, 8, 11, …
• 10, 20, 30, 40, …
• 6, 4, 2, 0, 02, …

## Algebraic Definition of Arithmetic Sequence

If $\{u_n\}$ is arithmetic, then $u_{n+1} – u_n = d$
for all positive integers $n$ where $d$ is a constant called the common difference.

If $a, \ b$ and $c$ are any consecutive terms of an arithmetic sequence then
\begin{aligned} \displaystyle \require{color} b – a &= c – b &\color{green} \text{equating common differences} \\ 2b &= a + c \\ \therefore b &= \frac{a+c}{2} \\ \end{aligned} \\
So, the middle term is the arithmetic mean of the terms on either side of it.

## The General Term Formula

Suppose the first term of an arithmetic sequence is $u_1$ and the common difference is $d$,
$u_n = u_1 + (n-1)d$

This formula can be referred to the following form as well.
$T_n = a + (n-1)d$
where the first term of an arithmetic sequence is $a$ and the common difference is $d$.

## Practice Questions of Arithmetic Sequence

### Question 1

Consider the arithmetic sequence, 4, 7, 10, 13, …, find a formula for the general term $u_n$.

### Question 2

Find the 100th term of an arithemtic sequence, 100, 97, 93, 89, …

### Question 3

Find $x$ given that $3x+1, \ x,$ and $-3$ are consecutive terms of an arithmetic sequence.

### Question 4

Find the general term $u_n$ for an arithmetic sequence with $u_4 = 3$ and $u_7 = -12$.

### Question 5

Insert four numbers between 3 and 23 so that all six numbers are in arithmetic sequence.