Question

How do you find the sum of the arithmetic sequence given 8, 3, -2, -7, -12, ....., -57?

Answer 1

Step 1: Find the number of terms

The nth term of an arithmetic sequence is given by `t_n = a + (n - 1)d`. We know `t_n` (-57), r (-5) and a (8), however we don't know n. We will therefore solve for `n`.

`-57 = 8 + (n- 1)-5`

`-57 = 8 - 5n + 5`

`-57 - 13 = -5n`

`-70 = -5n`

`n = 14`

`:.` The sequence has `14` terms.

Step 2: Apply the formula for sum:

When dealing with arithmetic series, we will frequently use two formulas to help us find the sum. They are:

•`s_n = n/2(2a + (n - 1)d)`

•`s_n = n/2(a + t_n)`

Since we know the first term, the last term and the number of terms, we have enough information to use the latter; this one is more efficient and easy to use than the first.

`s_n = n/2(a + t_n)`

`s_14 = 14/2(8 + (-57))`

`s_14 = 7(8 - 57)`

`s_14 = -343`

Hence, the sum of this arithmetic sequence is `-343`.

Practice exercises:

1. Determine the sum of the following sequence:

`3, 14, 25, ..., 179`

Hopefully this helps, and good luck!