Mock AIME 1 2007-2008 Problems/Problem 6
what the sigma
Solution
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 .
See also
Mock AIME 1 2007-2008 (Problems, Source) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |