Question

How do I find this summation with an unknown variable at the end? 8+7+6...+(9-1n).

Here's the question:
1. Find the sum:
8+7+6...+(9-1n).

I feel though I should use the formula `sum_(i=1)^nk` (with the "n" in the original question being the "i" in this summation formula). But I'm not sure what my upper limit would be. I also don't know how (or even if I should) solve for the variable in the question. At first, I thought that it was an infinite series and used a different formula, but it didn't work. Could someone help me? Thank you very, very much in advance!

Answer 1

`S_n=(n(17-n))/2=(17n-n^2)/2`

Explanation:

The sum of an arithmetic series ( a series which follows the sequence `a+bn`) can be found by doing `n/2[a+L]` where `n` is the number of terms, `a` is the first term, and `L` is the last term.

Here we have:
`n=n`
`a=8`
`L=9-n`

So, `S_n=n/2[8+9-n]`

`=n/2[17-n]`

`=(n(17-n))/2`

`=(17n-n^2)/2`

Given any value of `n, ninZZ^+`, the value of `S_n` can be found with the formula for that arithmetic series.