Question

Questions 48 to 50 are followed by two statements labelled as (1) and (2). You have to decide if these statements are sufficient to conclusively answer the question. Give answer:

(A) If statement (1) alone or statement (2) alone is sufficient to answer the question
(B) If you can get the answer from (1) and (2) together but neither alone is sufficient
(C) If statement 1 alone is sufficient to answer the question and statement (2) alone is also sufficient
(D) If neither statement (1) nor statement (2) is sufficient to answer the question

A sequence of numbers \( a_1, a_2, \ldots \) is given by the rule \( a_n^2 = a_{n+1} \). Does 3 appear in the sequence?

Statement 1: \( a_1 = 2 \).
Statement 2: \( a_3 = 16 \).

• A. If statement (1) alone or statement (2) alone is sufficient to answer the question
• B. If you can get the answer from (1) and (2) together but neither alone is sufficient
• C. If statement 1 alone is sufficient to answer the question and statement (2) alone is also sufficient
• D. If neither statement (1) nor statement (2) is sufficient to answer the question

Hint:

Work out the sequence forward from statement 1, and work backward carefully from statement 2, remembering a square cannot be negative; both pin the sequence down completely.

Answer 1

The Correct Option is C

Step 1: Understand the rule.
The rule \( a_n^2 = a_{n+1} \) means every next term is the square of the current term: \( a_2 = a_1^2 \), \( a_3 = a_2^2 \), \( a_4 = a_3^2 \), and so on. We want to know if the number 3 ever shows up anywhere in this chain.

Step 2: Test statement 1 alone: \( a_1 = 2 \).
Work out the sequence directly: \( a_1 = 2 \). \( a_2 = a_1^2 = 2^2 = 4 \). \( a_3 = a_2^2 = 4^2 = 16 \). \( a_4 = a_3^2 = 16^2 = 256 \). \( a_5 = a_4^2 = 256^2 = 65536 \), and so on, each term much bigger than the last.
Every term here is a whole power of 2, specifically \( 2^1, 2^2, 2^4, 2^8, 2^{16}, \ldots \), and the sequence only ever jumps from 2 straight to 4 to 16 to 256. It skips over 3 completely and never returns to a small value again, since squaring a number bigger than 1 always makes it bigger still. So the sequence never equals 3 at any point. This is a firm, complete no, so statement 1 alone is enough to answer the question.

Step 3: Test statement 2 alone: \( a_3 = 16 \).
Work backwards. Since \( a_3 = a_2^2 = 16 \), we get \( a_2 = 4 \) or \( a_2 = -4 \).
If \( a_2 = -4 \), then \( a_1^2 = a_2 = -4 \), which is impossible for a real number, since a square can never be negative. So \( a_2 = -4 \) cannot happen, leaving \( a_2 = 4 \) as the only possibility.
Now, \( a_1^2 = a_2 = 4 \), so \( a_1 = 2 \) or \( a_1 = -2 \). Either way, moving forward from \( a_2 = 4 \), the rest of the sequence is fixed: \( a_3 = 16 \), \( a_4 = 256 \), \( a_5 = 65536 \), and so on, exactly as before, and \( a_1 \) itself is either 2 or -2, neither of which is 3.
So regardless of whether \( a_1 \) is 2 or -2, none of the terms \( a_1, a_2, a_3, a_4, \ldots \) is ever equal to 3. Statement 2 alone also gives a firm no, so it too is enough by itself.

Step 4: Compare the two statements.
Both statement 1 and statement 2, when used completely separately from each other, are each enough on their own to conclude that 3 never appears in the sequence. Neither one needs the other.

Final Answer:
Statement 1 alone works, and statement 2 alone also works, independently of each other.
\[ \boxed{\text{Each statement alone is sufficient}} \]