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## Official Guide Explanation:

Data Sufficiency #107

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Quantitative Review (2nd ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

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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:

s_{1} = (1/1) - (1/2)

s_{2} = (1/2) - (1/3)

s_{3} = (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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