Arithmetic Sequences & Series

An arithmetic sequence is an ordered list of numbers in which the difference between any two consecutive terms is constant. This fixed difference is called the common difference, denoted $d$. If the first term is $u_1$, then every term is generated by repeatedly adding $d$: the sequence runs $u_1,\; u_1+d,\; u_1+2d,\; u_1+3d,\dots$. Because the growth is by equal steps, an arithmetic sequence models any situation of constant increase or decrease, such as a salary that rises by a fixed amount each year or the number of seats in successive rows of a theater.

The engine of the topic is the formula for the nth term (also called the general term): $$u_n = u_1 + (n-1)d.$$ The factor $(n-1)$ appears because you take $n-1$ steps of size $d$ to travel from the first term to the $n$th term. For exam