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.