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Migo1 Migo1 is offline
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Default Sum Formula with variable range

I need to calculate the moving average in column C for a long list of numbers
in Colum B. If the variable is in cell A1 and it = 3, then each sum formula
should sum a range of the rows, such as; Sum($B$1:$B3),
A1 is changed to 5, then Sum($B$1:$B5), where the variable in cell A1 is the
row number of end of the range.
How can this be accomplished?