CDS 2026 · PYQ · Data Sufficiency / Exponents · medium
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is (0.5)ⁿ + (0.5)⁻ⁿ always greater than 2, where n is an integer?
Statement-I: n is an even integer
Statement-II: n is negative
Which one of the following is correct in respect of the above Question and the Statements?
A.The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
B.The Question can be answered by using either Statement alone
C.The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
D.The Question can be answered even without using both the Statements✓ Correct
Explanation
Let a = (0.5)ⁿ. Then the expression is a + 1/a. For any positive real number a ≠ 1, a + 1/a > 2 by AM-GM inequality. The expression equals 2 only when a = 1, i.e., n = 0. So for any non-zero integer n, (0.5)ⁿ + (0.5)⁻ⁿ > 2. But if n = 0, the expression equals 1 + 1 = 2, which is not greater than 2. Since the question asks 'always greater than 2' for n being an integer, the answer is NO (because n = 0 is allowed and gives exactly 2). This conclusion can be reached without using either statement.
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