2006 iTest Problems/Problem 36
Let denote . The recursive sequence satisfies and, for all positive integers , Suppose that the series can be expressed uniquely as , where and are coprime positive integers and is not divisible by the square of any prime. Find the value of .
Solution
We write by rearranging the defining equation and using . Summing this with weights from zero to infinity, we get . We can rewrite this as .
Next, we compute, which simplifies to . Since , the entire expression becomes .
Taking square roots, we get , so our answer is and we are done.
See also
2006 iTest (Problems, Answer Key) | ||
Preceded by: Problem 35 |
Followed by: Problem 37 | |
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