# 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.