Some say that \(\displaystyle \sum_{k=1}^{k=n} k+\sum_{k=n}^{k=1} k =n+1\) and I don't understand this result!?

Well, "some" are wrong as you can easily check for yourself!

\(\displaystyle \sum_{k = 1}^5 k + \sum_{k = 1}^5 (5 - k + 1) = (1 + 2 + 3 + 4 + 5) + (5 + 4 + 3 + 2 + 1) = 30 \neq 5 + 1\)

-Dan

Addendum: Oh! I see what's happening. Notice that

\(\displaystyle \sum_{k = 1}^n k + \sum_{k = 1}^n (n - k + 1) = \sum_{k = 1}^n ( k + n - k + 1) = \sum_{k = 1}^n (n + 1)\)

\(\displaystyle \sum_{k = 1}^n k + \sum_{k = 1}^n (n - k + 1) = \sum_{k = 1}^n (n + 1)\)

So the statement you are confused about is written wrong... you are missing the summation on the RHS.

You really need to start writing \(\displaystyle \sum_{k = n}^{k = 1} k\) as the more standard \(\displaystyle \sum_{k = 1}^n (n - k + 1)\) which is the clue you need.