Let k be an even number. Is it possible to write 1 as the sum of the reciprocals of k odd integers?
The answer to the problem is no, and we will prove it by creating a contradiction when assuming that the above claim is true.
Assume for now that it is possible to write 1 as the sum of the reciprocals of an even number of odd integers.
In middle school arithmetic, we learned that . Generalized, suppose we want to evaluate , then letting
Now, because each of the ‘s are odd, then must also be odd. Furthermore, each of are also odd. We now rewrite the above equation as
Now, the left hand side is already shown to be odd. The right hand side however is the sum of also odd terms. However, because is an even number, the sum of an even number of odds becomes even. Therefore, if 1 can be written as the sum of the reciprocals of an even number of odd integers, then there exists some odd number than is equal to another even number. Because this creates a contradiction, then 1 cannot be expressed as above.