Q165 / 270 · NDA 2023 (Phase 1) · Mathematics

If $2^{\frac{1}{c}}, 2^{\frac{b}{a c}}, 2^{\frac{1}{a}}$ are in GP, then which one of the following is correct?

  • Correct answer
    A
    $a, b, c$ are in AP
  • B
    $a, b, c$ are in GP
  • C
    $a, b, c$ are in HP
  • D
    $a b, b c, c a$ are in AP

Solution

$2^{\frac{1}{c}}, 2^{\frac{b}{a c}}, 2^{\frac{1}{a}}$ are in GP $\left(2^{\frac{b}{a c}}\right)^2=2^{\frac{1}{c}} \cdot 2^{\frac{1}{a}}$ $2^{\frac{2 b}{a c}}=2^{\frac{1}{c}+\frac{1}{a}}$ $\therefore \quad \frac{2 b}{a c}=\frac{1}{c}+\frac{1}{a}=\frac{a+c}{a c}$ $\therefore \quad 2 b=a+c$ $\therefore \quad a, b, c$ are in AP.

Hence, the correct answer is option: 'A'.

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