Evaluate summation of negative quantity n minus 5 end quantity from n equals 2 to 8

One way of doing it is by using the properties of summing things, like factoring out the minus sign: ∑ n=2 8 - (n-5) = - ∑ n=2 8 (n-5)

and splitting it up into two parts and again factor = - (∑ n=2 8 n-5 ∑ n=2 8 1)

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=1 m n = m (m+1) 2

So what we get is = - (-1 + 8 (8+1) 2 - 7) = 8-72/2=8-36=-28.