Q112 / 155 · WBJEE 2016 · Mathematics

If the first and $(2 n-1)$ th terms of an $\mathrm{AP}$, GP and $\mathrm{HP}$ are equal and their $n$ th terms are respectively $a, b, c,$ then always

  • A
    $a=b=c$
  • Correct answer
    B
    $a \geq b \geq c$
  • C
    $a+c=b$
  • Correct answer
    D
    $a c-b^{2}=0$

Solution

$\because a, b$ and $c$ are respectively the $A M,$ GM and HM of first and $(2 n-1)$ th terms. and also, we know $A M \geq G M \geq H M$ and $A M \cdot H M=G M^{2}$ $\therefore \quad a \geq b \geq c$ and $a \cdot c=b^{2}$

Hence, the correct answer is options: 'B', 'D'.

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