Question: The sequence s1, s2, s3, ..., sn, ... is such that \(S_n\) = 1/n − 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 9/10?

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

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

The problem statement states that
Given:

• \(S_n\) = 1/n − 1/n + 1 for n ≥ 1.
• k is a positive integer

Find out:

• If the sum of the first k terms of the sequence is greater than 9/10.

Since \(S_n\) = 1/n − 1/n + 1, therefore we can say,
\(S_1\) = 1− ½
\(S_2\) = ½ - ⅓
\(S_3\) = ⅓ - ¼
…..

If we find the sum of the above 3 terms we will get:

\(S_1 + S_2 + S_3 \)=(1− ½) + (½ - ⅓) + (⅓ - ¼)

= 1− ¼ (everything except the first and the last numbers will cancel out).

Therefore, the sum of the first k terms is shown by the formula:

\(Sum_k\) =1− 1/k+1.

Hence, as per the question, we need to find out \(Sum_k\) =1− 1/k+1 > 9/10.
That is we can say, k/k+1 > 9/10
Or, we need to find out k > 9.

1. k > 10. Therefore the statement alone is sufficient.
2. k < 19. Therefore, the statement alone is not sufficient.

Approach Solution 2:

The problem statement suggests that
Given:

• \(S_n\) = 1/n − 1/n + 1 for n ≥ 1.
• k is a positive integer

Find out:

• If the sum of the first k terms of the sequence is greater than 9/10.

1. k > 10

Let the value of k is 11.

Therefore, Sum = \(S_1 + S_2 + S_3 + S_4 + S_5+ S_6 + S_7 + S_8 + S_9 + S_{10} + S_{11}\)

Where,

\(S_1\) = 1 - (1/2)
\(S_2\) = (1/2) - (1/3)
\(S_3\) = (1/3) - (1/4) …
\(S_{11}\) = (1/11) - (1/12)

That indicates, Sum = 1 - (1/12)
The terms like +1/2, -1/2, +1/3, and -1/3 will be added to zero. Only the first and last numbers remain.
=> Sum = 1 - 0.0XXXX > 9/10

Let’s take k as 12, then SUM = 1 - (1/13) which is > 0.9.
Hence, statement (1) alone is sufficient.

2. k < 19.

Let the value of k is 2.
Therefore, the sum is = 1 - (1/2) + (1/2) - (1/3) = 1 - (1/3) which is less than 9/10.
Let consider K = 11
Therefore, the sum is greater than 9/10 [already verified in (1) above]

Hence, statement (2) alone is not sufficient.

Approach Solution 3:
The problem statement informs that
Given:

• \(S_n\) = 1/n − 1/n + 1 for n ≥ 1.
• k is a positive integer

Find out:

• If the sum of the first k terms of the sequence is greater than 9/10.

First, it is required to find the pattern of the terms.
Let’s list the first few terms:

\(S_1\)= 1/1 −1/(1+1) = ½
\(S_2\)= ½ −1/(2+1) = ½ − ⅓ = ⅙
\(S_3\)= ⅓ − 1/(3+1) = ⅓ −1/4 = 1/12

Now, let’s find the pattern of the sum of terms:

\(S_1 + S_2\)= ½ + ⅙
= 4/6
= 2/3

\(S_1 + S_2 +S_3\) = ⅔ + 1/12

= 9/12
= 3/4

We can conclude that the sum of all terms up to the nth term will be n/(n+1).

Thus, as the question asks to find "if the sum of the first k terms of the sequence is greater than 9/10", we can summarise the question as:
"if the number of terms (n) is greater than 9"

Statement one alone:

k > 10
This implies that the number of terms in the sum is greater than 10.
Therefore, it is sufficient to answer the problem as Yes, it must be greater than 9.

Statement two alone:

k < 19
This does not inform us if the number of terms is greater than 9. The value of k could be 7 or it could be 11, etc.
Hence, Insufficient.

“The sequence s1, s2, s3, ..., sn, ... is such that \(S_n\) = 1/n − 1/n + 1”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Review". GMAT Quant section estimates a candidate’s knowledge to solve a sum mathematically. GMAT Data Sufficiency questions come up with a problem statement followed by two factual statements. GMAT Data Sufficiency includes 15 questions which are two-fifths of the whole 31 GMAT quant questions.

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