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.
The sequence S1, S2, S3..., Sn ... is such that
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- ceilidh.erickson
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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.
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.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
- MartyMurray
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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.VJesus12 wrote: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.
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.
Marty Murray
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Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.