# Everything You Need to Know about Arithmetic Sequences

An Arithmetic Sequence is a sequence in which each term differs from the previous one by the same fixed number. It can 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 Sequences

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 in 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, $4, 7, 10, 13, …$, and find a formula for the general term $u_n$.

\begin{aligned} \displaystyle u_1 &= 1 &\color{green} \text{the first term} \\ 7-4 &= 3 \\ 10-7 &= 3 \\ 13-10 &= 3 \\ d &= 3 &\color{green} \text{the common difference} \\ \therefore u_n &= 4 + (n-1) \times 3 \\ &= 4 + 3n-3 \\ &= 1 + 3n \end{aligned}

### Question 2

Find the $100$th term of an arithmetic sequence, $100, 97, 93, 89, …$.

\begin{aligned} \displaystyle u_1 &= 101 &\color{green} \text{the first term} \\ 97-101 &= -4 \\ 93-97 &= -4 \\ 89-93 &= -4 \\ d &= -4 &\color{green} \text{the common difference} \\ u_n &= 101 + (n-1) \times -4 \\ &= 101-4n + 4 \\ &= 105-4n \\ \therefore u_{100} &= 105-4 \times 100 \\ &= -295 \end{aligned}

### Question 3

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

\begin{aligned} \displaystyle x &= \frac{(3x+1)+(-3)}{2} \\ 2x &= 3x-2 \\ \therefore x &= 2 \end{aligned}

### Question 4

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

\begin{aligned} \displaystyle \require{AMSsymbols} u_4 &= u_1 + 3d = 3 \color{green} \cdots \text{(1)} \\ u_7 &= u_1 + 6d = -12 \color{green} \cdots \text{(2)} \\ (1)-(2) \\ -3d &= 15 \\ d &= -5 \\ u_1 + 3 \times (-5) &= 3 &\color{green} \text{substitute } d=-5 \text{ into (1)} \\ u_1 &= 18 \\ u_n &= 18 + (n-1) \times (-5) \\ &= 18 -5n + 5 \\ \therefore u_n &= 23-5n \end{aligned}

### Question 5

Insert four numbers between $3$ and $23$, so all six numbers are in an arithmetic sequence.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \text{Six numbers are } 3, \ 3+d, \ 3+2d, \ 3+3d, \ 3+4d, \ 3+5d. &\color{green} \ \ d \text{ is the common difference} \\ 3 + 5d &= 23 \\ 5d &= 20 \\ d &= 4 \\ 3+4, \ 3+2 \times 4, \ 3+3 \times 4, \ 3+4 \times 4 \\ \therefore 7, \ 11, \ 15, \ 19 \end{aligned} 