Everything You Need to Know about Arithmetic Sequences

Arithmetic Sequence

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 \( d \).

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Practice Questions of Arithmetic Sequence

Question 1

Consider the arithmetic sequence, \( 4, 7, 10, 13, … \), and find a formula for the general term \( u_n \).

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

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Question 2

Find the \(100\)th term of an arithmetic sequence, \( 100, 97, 93, 89, …\).

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

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Question 3

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

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

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Question 4

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

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

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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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