Mock AIME 1 2007-2008 Problems/Problem 6
A -array is a structured, infinite, collection of numbers. For example, a -array is constructed as follows:
In general, the first entry of each row is times the first entry of the previous row. Then, each succeeding term in a row is times the previous term in the same row. If the sum of all the terms in a -array can be written in the form , where and are relatively prime positive integers, find the remainder when is divided by .
Note that the value in the th row and the th column is given by . We wish to evaluate the summation over all , and so the summation will be, using the formula for an infinite geometric series: Taking the denominator with (indeed, the answer is independent of the value of ), we have (or consider FOILing). The answer is .
With less notation, the above solution is equivalent to considering the product of the geometric series . Note that when we expand this product, the terms cover all of the elements of the array.
By the geometric series formula, the first series evaluates to be . The second series evaluates to be . Their product is , from which we find that leaves a residue of upon division by .
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