1 − 2 2 1 + 1 − 4 2 1 + 1 − 6 2 1 + 1 − 8 2 1 + ⋯ + 1 − 1 0 0 0 2 1
The value of the expression above can be expressed as b a , where a and b are coprime integers.
What is the number of ways (equivalently, the number of ordered pair of integers ( a , b ) ) the value of the expression above can be expressed as b a , where a and b are coprime integers.
Hint : Is it at all needed to compute the sum?
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For the record, the sum, (evaluated using telescoping series), is − 1 0 0 1 5 0 0 .
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Clearly, every term of the series is negative. So, the sum will be negative too.
Now, if the sum is equal to b a , with a and b are integers, then exactly one of a and b has to be negative.
As a and b are coprime, we have only one option for ( ∣ a ∣ , ∣ b ∣ ) , which yields 2 options for ( a , b ) : either ( − ∣ a ∣ , ∣ b ∣ ) or ( ∣ a ∣ , − ∣ b ∣ ) .