For a general arithmetic sequence with first term \(a\) and a common difference \(d\), we can generate the following terms:
\begin{align*} {T}_{1}&= a \\ {T}_{2}&= {T}_{1}+d = a + d \\ {T}_{3}&= {T}_{2}+d = \left(a+d\right)+d =a+2d \\ {T}_{4}&= {T}_{3}+d=\left(a+2d\right)+d=a+3d \\ \vdots & \quad \quad \qquad \vdots \quad \quad \qquad \vdots \quad \quad \qquad \vdots \\ {T}_{n}&= {T}_{n1} + d =\left(a+(n2)d\right)+d = a + \left(n1\right)d \end{align*}
Therefore, the general formula for the \(n\)\(^{\text{th}}\) term of an arithmetic sequence is:
\[{T}_{n}= a + \left(n1\right)d\]
Arithmetic sequence
An arithmetic (or linear) sequence is an ordered set of numbers (called terms) in which each new term is calculated by adding a constant value to the previous term:
\[{T}_{n}=a+(n1)d\]
where

\({T}_{n}\) is the \(n\)\(^{\text{th}}\) term;

\(n\) is the position of the term in the sequence;

\(a\) is the first term;

\(d\) is the common difference.
Test for an arithmetic sequence
To test whether a sequence is an arithmetic sequence or not, check if the difference between any two consecutive terms is constant:
\[d = {T}_{2}{T}_{1}={T}_{3}{T}_{2}= \ldots = {T}_{n}{T}_{n1}\]
If this is not true, then the sequence is not an arithmetic sequence.
Worked example 1: Arithmetic sequence
Given the sequence \(15; 11; 7; \ldots 173\).
 Is this an arithmetic sequence?
 Find the formula of the general term.
 Determine the number of terms in the sequence.
Check if there is a common difference between successive terms
\begin{align*} T_{2}  T_{1} &= 11  (15) = 4 \\ T_{3}  T_{2} &= 7  (11) = 4 \\ \therefore \text{This is an } & \text{arithmetic sequence with } d = 4 \end{align*}
Determine the formula for the general term
Write down the formula and the known values:
\[T_{n} = a + (n1)d\] \[a = 15; \qquad d = 4\] \begin{align*} T_{n} &= a + (n1)d \\ &= 15 + (n1)(4) \\ &= 15 + 4n  4 \\ &= 4n  19 \end{align*}
A graph was not required for this question but it has been included to show that the points of the arithmetic sequence lie in a straight line.
Note: The numbers of the sequence are natural numbers \(\left(n \in \{1;2;3; \ldots \} \right)\) and therefore we should not connect the plotted points. In the diagram above, a dotted line has been used to show that the graph of the sequence lies on a straight line.
Determine the number of terms in the sequence
\begin{align*} T_{n} &= a + (n1)d \\ 173 &= 4n  19 \\ 192 &= 4n \\ \therefore n &= \frac{192}{4} \\ &= 48 \\ \therefore T_{48} &= 173 \end{align*}
Write the final answer
Therefore, there are \(\text{48}\) terms in the sequence.