\(S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}\)
GMAT quant questions tend to be designed to be answered efficiently via the use of hacks. So, let's start hacking.
Start off with the obvious: \(1 + \frac{1}{2^2} = 1 \frac{1}{4}\)
It becomes apparent that to get to 2 we need 3/4 more, and those other fractions look pretty small. They probably won't get us to 2.
If we can prove that 1 1/4 + (the sum of the rest of those fractions) < 2, then the correct answer will be S < 2 and we'll be done.
The next fraction is \(\frac{1}{3^2}\).
That is effectively \(\frac{1}{9}\).
We have 1 1/4 + 1/9.
Without adding, we know that 1 1/4 + 1/9 < 1 1/2.
The next seven are \(\frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}\).
Since 8 x 1/16 = 1/2, and the first fraction of this set of seven fractions is 1/16 and all the rest are smaller, this set has to add up to less than 1/2.
So, S = 1 1/4 + 1/9 + (a number < 1/2).
S < 2.
The correct answer is (E).
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