The sequence S1, S2, S3..., Sn ... is such that

[VJesus12]

The sequence S1, S2, S3..., Sn ... is such that $$S_n=\frac{1}{n}-\frac{1}{n+1} .$$ If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?

(1) k > 10
(2) k < 19

The OA is the option A.

If the first statement is sufficient, why is not sufficient the second one? Could anyone give me a good explanation here? Please.

[ceilidh.erickson]

Whenever a question asks about the SUM of a SEQUENCE, we have to find patterns.

First, we need to find the pattern of the terms. List out the first few terms:

$$S_1=\frac{1}{1}-\frac{1}{1+1}=\frac{1}{2}$$
$$S_2=\frac{1}{2}-\frac{1}{2+1}=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$$
$$S_3=\frac{1}{3}-\frac{1}{3+1}=\frac{1}{3}-\frac{1}{4}=\frac{1}{12}$$

Now find the pattern of the sum of terms:
$$S_1+S_2=\frac{1}{2}+\frac{1}{6}=\frac{4}{6}=\frac{2}{3}$$
$$S_1+S_2+S_3=\frac{2}{3}+\frac{1}{12}=\frac{9}{12}=\frac{3}{4}$$
We can infer that the sum of all terms up to the nth term will be
$$\frac{n}{n+1}$$
Thus, if the question is asking "is the sum of the first k terms of the sequence greater than 9/10?", we can rephrase the question as:
"is the number of terms (n) greater than 9?"

(1) k > 10
This tells us that the number of terms in our sum is greater than 10. This is sufficient to tell us that yes, it must be greater than 9.

(2) k < 19
This does not tell us whether the number of terms is greater than 9. k could be 7 or it could be 11, etc. Insufficient.

The answer is A.

[MartyMurray]

The sequence S1, S2, S3..., Sn ... is such that $$S_n=\frac{1}{n}-\frac{1}{n+1} .$$ If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?

(1) k > 10
(2) k < 19

The OA is the option A.

If the first statement is sufficient, why is not sufficient the second one? Could anyone give me a good explanation here? Please.

A general lesson from this example is that, while one simple constraint, in this case k > 10, may be enough to define the answer to a question, another similar constraint, in this case, K < 19, may not be sufficient.

The GMAT often uses such constraints in the answers to DS questions. In this case, k > 1 would not be sufficient either, while either k < 5 or k > 20 would be sufficient.

For further practice, you could determine why k > 1 is not sufficient and why k < 5 and k > 20 are sufficient.