Sum a variable number of cells

Essentially, I would like to have a cell in A2 sum up the values in cells A1, B1, C1 where the number of cells it sums is a variable in another cell that may change. I would like the same to apply to cell B2, C2, D2, etc.

This may make more sense if you look at the attachment.

Thanks! This puzzle has been bugging me for a while.