Q89 / 180 · KCET 2018 · Mathematics

If \( a, b, c \) are three consecutive terms of an \( A P \) and \( x, y, z \) are three consecutive terms of a G.P.,
then thevalue of \( X^{b-c}, Y^{c-a}, Z^{a-b} \)
is

  • A
    \( 0 \)
  • B
    \( X Y Z \)
  • C
    \( -1 \)
  • Correct answer
    D
    \( 1 \)

Solution

Given that, $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. and $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in G.P. Let $\mathrm{d}$ be the common difference between A.P. consecutive terms. So, $b-a=c-b=d, c-a=2 d$ Now, $\chi^{b-c} \cdot y^{c-a} \cdot z^{a-b}$ $=x^{-d} \cdot y^{2 d} \cdot z^{-d}=(x z)^{-d} \cdot\left(y^{2}\right)^{-d}$ Since, $x z=y^{2}$ then $(x z)^{-d}(x z)^{d}=1$

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

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