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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Consider the sum:

equation

Which of the following statements is true?

Perform the first subtraction and then factor to obtain:

equation

equation

By repeating this operation, we get:

equation

equation

equation

and, if we continue like this for about 4999 times, we get:

equation

and, therefore, S is greater than zero but smaller than one half.

Only III is true.