Arithmetic Progressions

What is an Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference ($d$). For example: $2, 5, 8, 11, 14, \ldots$ is an AP with first term $a = 2$ and common difference $d = 3$.

Formally, $a_1, a_2, a_3, \ldots$ is an AP if $a_{n+1} - a_n = d$ for all $n \geq 1$.

The General (nth) Term

The $n^{\text{th}}$ term of an AP with first term $a$ and common difference $d$ is:
$$a_n = a + (n-1)d$$

Applications: find the $n^{\text{th}}$ term, check if a number belongs to an AP, find which term has a given value, find the number of terms in a finite AP.

Sum of First n Terms

$$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)$$
where $l = a_n$ is the last term. The second form is useful when the last term is known.

Important relation: $a_n = S_n - S_{n-1}$ for $n \geq 2$, and $a_1 = S_1$.