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.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
