GMAT考题：OG2017Q-DS-285

GMAT考题

题库 / OG2017Q-DS-285

The sequence s₁, s₂, s₃, ..., s_n, ... is such that ~$s_n=\frac{1}{n}-\frac{1}{n+1}$~for all integers n ≥ 1. If K is a positive integer, is the sum of the first k terms of the sequence greater than ~$\frac{9}{10}$~ ?

(1) k > 10

(2) k < 19

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

考题讲解

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题目分析：

前k项的和为~$(\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+……+(\frac{1}{k}-\frac{1}{k+1})=1-\frac{1}{k+1}=\frac{k}{k+1}$~，即求和过程中中间项的前半部分和前一项的后半部分相互抵消。

要使得前k项的和大于~$\frac{9}{10}$~，则~$\frac{k}{k+1}＞\frac{9}{10}$~，解得k>9。

（1）条件1给出k>10，满足k>9的条件，能够判断前k项的和大于~$\frac{9}{10}$~，题目得解。

（2）条件2给出k<19，也不能保证k>9，因此也不能判断前k项的和是否大于~$\frac{9}{10}$~。即若9<k<19，则前k项的和大于~$\frac{9}{10}$~，否则前k项的和小于~$\frac{9}{10}$~。条件2不满足要求。

因此，本题答案为（A），仅条件1可以解题，条件2不行。

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