Why is $\sum_{k=j}^{i+j}(j+i-k) = \sum_{k=1}^{i}(k)$ $\displaystyle\sum_{k=j}^{i+j}(j+i-k) = \displaystyle\sum_{k=1}^{i}(k)$
I know the above are equal through testing it out with arbitrary values, but I can't get an intuitive grasp as to why this is. 

Write out the terms in the first sum:
$$\sum_{k=j}^{i+j} (j+i-k)=i+(i-1)+\cdots + 1 + 0 = \sum_{k=1}^i k$$
