Question

Is \( 3^n > 2^k \)? Statement (I): \( k = n + 1 \)
Statement (II): \( n \) is a positive integer.

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

When an inequality depends on a variable, always test the smallest possible values (like \( n=1, 2 \)). If the result changes, the statement is insufficient.

Answer 1

The Correct Option is D

Concept: This is an inequality problem involving exponents. We need to determine if the relationship \( 3^n > 2^k \) holds true.

Step 1: Analyze Statement (I). Substituting \( k = n + 1 \) into the inequality: \[ 3^n > 2^{n+1} \] We can compare these by dividing both sides by \( 2^n \): \[ \frac{3^n}{2^n} > \frac{2^{n+1}}{2^n} \implies \left( \frac{3}{2} \right)^n > 2 \] Or simply testing values: If \( n=1 \): \( 3^1 > 2^2 \implies 3 > 4 \) (False). If \( n=2 \): \( 3^2 > 2^3 \implies 9 > 8 \) (True). Since the inequality holds for some values of \( n \) and not others, Statement (I) is insufficient.

Step 2: Analyze Statement (II). Knowing \( n \) is a positive integer does not provide the value of \( k \) or any relationship to \( n \). Therefore, Statement (II) is insufficient.

Step 3: Analyze both statements together. Even with both statements, we know \( n \) is a positive integer and \( k = n + 1 \). As demonstrated in Step 1, the inequality \( 3^n > 2^{n+1} \) is false for \( n=1 \) and true for \( n \ge 2 \). Since the result depends on the specific value of \( n \), both statements combined remain insufficient. Neither statement is sufficient.