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!

1 Answer
Jan 12, 2018

#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.