Mock AIME 1 2007-2008 Problems/Problem 6
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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 
.
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 | ||
