I'm stuck on this summation problem $$\sum_{k=0}^4 (-1)^k\sum_{l=k+1}^5l$$

I can solve both of the summations, but I don't know what to do when they're next to each other like that, the answer should be $9$ and I have no idea how to get to that. Any help would be much appreciated, thank you very much


closed as off-topic by Henrik, Mark, Namaste, Don Thousand, ArsenBerk Oct 19 '18 at 22:09

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    $\begingroup$ As $k$ (the variable the first sum is over) is used in the second sum, it's an inner sum (i.e. there could - some would say should, for clarity - have been parentheses around $(-1)^k$ and the second sum). As you say you can do that sum, do so, then you'll have one sum left. $\endgroup$ – Henrik Oct 19 '18 at 19:14
  • $\begingroup$ Thank you very much, I think I overestimated myself (with knowing how to do it) because even so, I still can't get to the number 9. I think it would help me to see the steps, because I'm starting to feel hopeless $\endgroup$ – G.Jan Oct 19 '18 at 19:46

The second sum is part of the first sum. For each $k$ from $0$ to $4$, you are meant to evaluate $(-1)^k\sum_{l=k+1}^5 l$. Then you take those five results and add them.

  • $\begingroup$ Thank you! okay, I understand that, but still I'm too confused, I always end up with 1, -2, 3, -4, 5 but that gives 3, would you mind writing the steps for me (it's okay if you don't, I just can't figure it out on my own) $\endgroup$ – G.Jan Oct 19 '18 at 19:25
  • $\begingroup$ @G.Jan $(-1)^k\sum_{l=k+1}^5 l$ for $k=0$ becomes $(-1)^0\sum_{l=1}^5 l=1\cdot(1+2+3+4+5)=15$. Then for $k=1$ we get ... $\endgroup$ – Arthur Oct 19 '18 at 20:35
  • $\begingroup$ Thank you so very much, I finally got it and I feel so stupid, I really needed a push, so thank you ! $\endgroup$ – G.Jan Oct 19 '18 at 22:17

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