One thing that bothers me about this is you say "The Last Term (An) Is 0.5 = 31/16". Presumably you are aware the 0.5 is NOT equal to 21/16 so that cannot mean what you say. What did you mean by that? Is it possible that the first term is 1, the "common ratio" is 0.5 and the sum of the first M terms is 31/16?
If that is the case then you need to solve for M.