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: Understanding the Concept:
The rule says each term is the square of the one before it. Once a term is bigger than 1, squaring it only makes it grow, and it grows very fast. This growth idea is the key to answering both statements without listing every term by hand.

Step 2: Key Formula or Approach:
If any term of the sequence is a whole number bigger than 1, then every later term is also a whole number, and the terms strictly increase term after term, since squaring a number greater than 1 gives an even bigger number. So once we know one term exactly, and that term is a whole number greater than 1, we can be sure the sequence never lands back on a smaller specific number like 3, because it only moves upward from there, and we can check the handful of terms below it directly.

Step 3: Detailed Explanation:
Statement 1 gives $a_1 = 2$. Squaring repeatedly: $a_2 = 4$, $a_3 = 16$, $a_4 = 256$, each one a perfect power of 2 climbing fast. None of $2, 4, 16, 256, \ldots$ equals 3, and the sequence keeps growing forever after, so 3 can never turn up later either. Statement 1 alone settles the question with a clear no.
Statement 2 gives $a_3 = 16$. We do not know $a_1$ directly, but we can pin it down. Since $a_2^2 = 16$, $a_2$ is 4 or -4; but $a_2$ itself must be a square, since $a_2 = a_1^2$, and a square can never be negative, so $a_2 = 4$ is the only option. Then $a_1^2 = 4$ gives $a_1 = 2$ or $a_1 = -2$.
In both cases, $a_1$ is 2 or -2, and from $a_2 = 4$ onward the sequence is identical to the one built from statement 1: 4, 16, 256, and so on, growing without bound. Checking the only terms that could possibly be small, $a_1$, which is 2 or -2, shows neither is 3, and every term from $a_2$ onward is 4 or larger and only increasing, so 3 is ruled out everywhere. Statement 2 alone also settles the question with a clear no.

Step 4: Final Answer:
Both statements, used completely separately, are each able to prove on their own that 3 never appears in the sequence, so both statement (1) alone and statement (2) alone are sufficient.
\[ \boxed{\text{Each statement alone is sufficient}} \]