Question

How do you find the sum of the arithmetic series -12-9-6-...+39?

Answer 1

There are 2 formulae to do this, but you need to find how many terms there are first.

`"S_18 = 243`

Explanation:

The sum of an Arithmetic series can be found from:

`S_n = n/2[2a + (n-1)d]" "` or `" "S_n = n/2(a + l)`

However both of these require knowing how many terms there are.

Each term is defined by `T_n = a + (n - 1)d`

What do we know already?
`a = -12, " " d = 3, " " T_n = 39`

`T_n = (-12) + (n-1)(3) = 39" solve to find n"`

`-12 + 3n - 3 = 39`
`3n = 39 + 15`
`3n = 54`
`n = 18 " there are 18 terms"`

Now we can use either formula for the sum of the first 18 terms.
`S_n = n/2[2a + (n-1)d]" "` or `" "S_n = n/2(a + l)`

`S_18 = 18/2[2(-12) + (18-1)3] "` or `" "S_18 = 18/2(-12 + 39)`

`S_18 = 9[-24 + 51]" "` or `" "S_18 = 9(27)`
`S_18 = 9 xx 27" "` or `" "S_18 = 243`
`"S_18 = 243`

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