Official Guide Explanation:
Data Sufficiency #107

Background

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Solution and Metadata

Question: 107
Page: 160
Difficulty: 7 (Very Difficult)
Category 1: Algebra > Functions > Sequences
Category 2: Arithmetic > Fractions >

Explanation: First, tinker with the sequence to learn that it isn't quite as complicated as it looks at first glance:

s₁ = (1/1) - (1/2)

s₂ = (1/2) - (1/3)

s₃ = (1/3) - (1/4)

The question asks about the sum of the first k terms. Note that if we add up these terms, many of the component parts cancel out. For instance, if we take the first three terms:

sum = ((1/1) - (1/2)) + ((1/2) - (1/3)) + ((1/3) - (1/4))

The -(1/2) and (1/2) cancel out, as well as the (1/3)s, leaving:

sum = (1/1) - (1/4)

If k were a much larger number, such as 10, we could apply the same principle:

sum = (1/1) - (1/11)

In other words, the sum will always be:

sum = 1 - (1/(k + 1))

It will be greater than (9/10) if and only if k is greater than 9. If k = 9, then the sum is equal to (9/10), and if it's less, the sum is less still.

Statement (1) is sufficient. If k > 10, then the sum is no less than 1 - (1/12) = (11/12), which is greater than (9/10).

Statement (2) is insufficient. If k < 19, it could be greater than 9, which gives us a sum greater than (9/10), but it could be much smaller. If k = 2, then the sum is 1 - (1/3) = (2/3). Choice (A) is correct.

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