Question

If a, b, c are the three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P., then prove that x^{b – c}. y^{c – a}. z^{a – b} = 1.

Answer 1

a, b, c are in A.P. ⇒ 2b = a + c                  ...(i) 

x, y, z are in G.P. ⇒ y = xr, z = xr², where r is the common ratio 

∴ x^{b – c}. y^{c – a}. z^{a – b} = x^{b – c}. (xr)^{c – a}. (xr²)^{a – b} 

= x^{b – c} . x^{c – a} . r^{c – a}. x^{a – b} . r^{2a – 2b} = x^{b – c + c – a + a – b}. r^{c – a + 2a – 2b} 

= x⁰ . r^{c + a – 2b} = x⁰.r^{2b – 2b} = x⁰.r0 = 1    (From (i))