Arithmetic Sequences as Linear Recurrences

An arithmetic sequence is a sequence where each successive term differs by a constant amount. The arithmetic sequence resembles the sequence: $a_n = a_{n-1} + d$ where $a_1$ is the first element and each term afterwards has the common difference term $d$ added to it, resulting in $a_1, a_1+d, a_1+2d...$ and to generalize, the nth term of this sequence is $a_n = a_1 + (n-1)d$.

An example would be the natural numbers where $a_1 =1, d= 1$ resulting in the progression: $1,2,3...$

Properties: $$S_n = \frac{n(a_1+a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}$$ The average of the elements in an arithmetic sequence is $$\bar{n} = \frac{(a_1+a_n)}{2}$$