Question

If the sequence S has 100 terms, what is the 7th term of S?
(1) The 93rd term of S is -100, and each term of S after the first is 7 less than the preceding term.
(2) The first term of S is 544.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

In data sufficiency questions about sequences, you need two pieces of information: the type of sequence (e.g., arithmetic, geometric) and a reference point (e.g., the value of a specific term). Statement (1) provides both, while statement (2) only provides a reference point.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks for a specific term (the 7th) of a sequence. To find this, we need to know the rule that governs the sequence and at least one of its terms.
Step 2: Key Formula or Approach:
Statement (1) describes an arithmetic progression, which is a sequence where the difference between consecutive terms is constant. The formula for the \(n\)-th term of an arithmetic progression is:
\[ a_n = a_m + (n-m)d \]
where \(a_n\) is the \(n\)-th term, \(a_m\) is the \(m\)-th term, and \(d\) is the common difference.
Step 3: Detailed Explanation:
Analyzing Statement (1):
This statement provides two key pieces of information:
1. The sequence is an arithmetic progression because "each term of S after the first is 7 less than the preceding term." This means the common difference, \(d\), is -7.
2. The 93rd term, \(a_{93}\), is -100.
We want to find the 7th term, \(a_7\). We can use the formula with \(n=7\), \(m=93\), \(d=-7\), and \(a_{93}=-100\).
\[ a_7 = a_{93} + (7-93)d \]
\[ a_7 = -100 + (-86)(-7) \]
\[ a_7 = -100 + 602 \]
\[ a_7 = 502 \]
Since we can find a unique value for the 7th term, statement (1) alone is sufficient.
Analyzing Statement (2):
This statement tells us that the first term, \(a_1\), is 544.
However, this statement alone does not provide the rule for the sequence. We do not know if it is an arithmetic sequence, a geometric sequence, or something else. Without the common difference or another rule, we cannot determine the 7th term from the first term alone. Therefore, statement (2) alone is not sufficient.
Step 4: Final Answer:
Statement (1) is sufficient to answer the question, but statement (2) is not. Therefore, the correct option is (A).