First, let's expand the expression inside the summation:
$$2+i-i^{2}$$
Now, let's substitute the values of $i$ from 1 to 6 into the expression:
$$\begin{align*}
\text{When } i=1: & \quad 2+1-1^{2}=2+1-1=2+1-1=2 \\
\text{When } i=2: & \quad 2+2-2^{2}=2+2-4=4
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