Arithmetic Sequences

An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic sequence with common difference of $2$.

• $a_1$: The first term of the sequence
• $d$: The common difference of successive terms
• $a_n$: The $n$th term of the sequence

The behavior of the arithmetic sequence depends on the common difference $d$.

If the common difference, $d$, is:

• Positive, the sequence will progress towards infinity ($+\infty$)
• Negative, the sequence will regress towards negative infinity ($-\infty$)

Note that the first term in the sequence can be thought of as $a_1+0\cdot d,$ the second term can be thought of as $a_1+1\cdot d,$ the third term can be thought of as $a_1+2\cdot d, $and so the following equation gives $a_n$:

$a_n= ​a_1+(n−1) \cdot d$

Of course, one can always write out each term until getting the term sought—but if the 50th term is needed, doing so can be cumbersome.