Arithmetic progression

Example

Let $a$ and $d$ be real numbers. The sequence $(a_n)_{n \ge 0}$ with $a_n = a + n d$ is the arithmetic progression with first term $a$ and common difference $d$. It is a sequence in which each term minus the previous one is the same number: $a_{n+1} - a_n = d$ for every $n \ge 0$. For example, $a = 3$ and $d = 4$ give $3, 7, 11, 15, \ldots$.

The sum of the first $n$ terms is $$a_0 + a_1 + \cdots + a_{n-1} = n a + \frac{n(n-1)}{2} d = \frac{n (a_0 + a_{n-1})}{2}.$$ Check: $\sum_{k=0}^{n-1} (a + kd) = na + d \sum_{k=0}^{n-1} k$, and $\sum_{k=0}^{n-1} k = n(n-1)/2$ by the sum of the first natural numbers. For $3 + 7 + 11 + 15$ this gives $4 \cdot (3 + 15) / 2 = 36$.