CDS 2026 (I) Elementary Mathematics, question 78

Topic: Number system

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Q78

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 StatementsCorrect answer

Solution.

Answer: (d) The Question can be answered even without using both the Statements

Let $t = 2^{n} > 0$ . Then $(0.5)^{n} + (0.5)^{-n} = t + \frac{1}{t}$ and $t + \frac{1}{t} - 2 = \frac{(t - 1)^{2}}{t} \geq 0$ .
Equality holds only when $t = 1$ , that is $n = 0$ ; for every other integer the sum is greater than 2.
Since $n = 0$ is an integer, the sum is not always greater than 2, so the Question is answered No from its own condition, without either Statement.

Trap: Statement II alone also gives a definite answer (Yes, for negative n), which makes (a) or (b) tempting, but the Question asks always for any integer n and is settled before the Statements are used.