One way of doing it is by using the properties of summing things, like factoring out the minus sign: ∑n=28−(n−5)=−∑n=28(n−5)
and splitting it up into two parts and again factor =−(∑n=28n−5∑n=281)
The last sum is just counting how many numbers there are from 2 to 8, which is 7 (you can do it on your fingers). The first one can be calculated fast if we are clever and use this identity:
∑n=1mn=m(m+1)2
So what we get is =−(−1+8(8+1)2−7)=8−72/2=8−36=−28.