Arithmetic sequences are characterized by the fact that to get from one term to the next we always add the same amount.
The amount we add is known as the common difference and is usually referred to as \(d\).
For example, the sequence whose first few terms are:
\[3,7,11,15,19,23, \dots \]
is arithmetic, with common difference \(d = 4\).
Given the first few terms of an arithmetic sequence, we can calculate any term of the sequence, using formula:
\[u_n = u_1 + \begin{pmatrix} n -1 \end{pmatrix}d \]
Where \(u_1\) is the first term of the sequence and \(d\) is its common difference.
This formula can also be rearranged and written:
\[u_n = dn+c\]
Where \(c\) is a number that we'll quikcly know how to find.
These formula allow us to calculate any term of the sequence directly.
In the following tutorial we review the method for findng the formula for the \(n^{\text{th}}\) term of a linear sequence. Watch it now.
Given an arithmetic sequence, we can calculate the sum of its first \(n\) terms, \(S_n\), using the formula:
\[S_n = \frac{n}{2} \begin{pmatrix} 2.u_1 + (n-1)d \end{pmatrix}\]
Where \(u_1\) is the first term of the sequence and \(d\) is its common difference.
Note: exam questions frequently involve this second formula.