Arithmetic sum

The arithmetic sum of a sequence of numbers is

$$\frac{n(a_1+a_n)}{2}$$

where $a_1$ is the smallest value in the sequence and $a_n$ is the biggest value in the sequence.

Geometric sum

When $r\ne 1$, the sum of the first $n+1$ terms is

$$a+ar+a{r}^2+a{r}^3+\cdots +a{r}^n=\sum _{k=0}^{n}a{r}^k=a\left(\frac{1-r^{n+1}}{1-r}\right)$$

Geometric sum for probabilities

When $r$ is between $[0,1]$ we can write the geometric sum as

$$\sum _{k=0}^{\infty }a{r}^k=\frac{a}{(1-r)}$$