manalq8, that approach only works for consecutive integers, because they are evenly spaced. Since the consecutive integers are in the denominator here, the terms are not evenly spaced (i.e. the difference between 1/100 and 1/101 is not the same as the difference between 1/101 and 1/102). As joeshmo pointed out, the averaging approach would not get us a precise answer for this problem.
To illustrate, let's look at a simpler set of fractions:
1/1 + 1/2 + 1/3 =
If we multiply the median by the number of terms, we'd say the total is 3/2. However, the actual total is 11/6, which is about 22% greater than the projected answer. Also, notice that in this case, the average of the first and last terms does not equal the median. (1/1 + 1/3)/2 = (4/3)/2=4/6 = 2/3 If we average all 3 terms, we get yet another result (11/18). You can try this with any set of fractions and you'll get similar results. The consecutive numebr tricks are great, but they only work if the actual terms in question are evenly spaced!
Dmitry Farber | Manhattan GMAT Instructor | New York
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