Question

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

Answer 1

`680`

Explanation:

The sum of a series is

`n/2["first" + "last"]`,

where `n` is the number of terms (in this case 20), `"first"` is the first term in the series and `"last"` is the last one.

Substituting in our values of `"first" = 81, "last" = -13, n = 20`

`10[81 - 13] = 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 = oo`, 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 `-oo`.