# How do you find the sum of the arithmetic sequence having the data given a_1=81, a_n = - 13, n = 20?

Apr 5, 2016

$680$

#### Explanation:

The sum of a series is

$\frac{n}{2} \left[\text{first" + "last}\right]$,

where $n$ is the number of terms (in this case 20), $\text{first}$ is the first term in the series and $\text{last}$ is the last one.

Substituting in our values of $\text{first" = 81, "last} = - 13 , n = 20$

$10 \left[81 - 13\right] = 680$

which is the sum of the series between $n = 1$ and $n = 20$.

If you wanted to find the total sum between $n = 1$ and $n = \infty$, you can solve this one using mere common sense.

The series clearly has a decreasing common difference if the first is $81$ and the 20th is $- 13$, and so there will be in total more terms that are negative than positive, as everything after $- 13$ will also be negative. Therefore, adding them all together, you will approach $- \infty$.