Question

If $n$ and $k$ are positive integers, is $\sqrt{n+k}>2\sqrt{n}$?
(1) $k>3n$
(2) $n+k>3n$

• 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.

Answer 1

The correct option is A Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

• Determine if . Since each side is positive, squaring each side preserves the inequality, so is equivalent to , which in turn is equivalent to $n+k>4n$, or to $k>3n$.

1. Given that $k>3n$, then ; SUFFICIENT.
2. Given that $n+k>3n$, then $k>2n$. However, it is possible for $k>2n$ to be true and $k>3n$ to be false (for example, $k=3$ and $n=1$) and it is possible for $k>2n$ to be true and $k>3n$ to be true (for example, $k=4$ and $n=1$); NOT sufficient.

• The correct answer is A; statement 1 alone is sufficient.