# 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,

- \( 2, 5, 8, 11, \cdots \)
- \( 10, 20, 30, 40, \cdots \)
- \( 6, 4, 2, 0, -2, \cdots \)

## 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} \)

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