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.

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

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