Question

If a, b, c is 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 is three consecutive terms of an AP.

∴ Let a, b, c be a, a + d, a + 2d respectively  ...(1)

x, y, z is three consecutive terms of a GP.

∴ Assume x, y, z as x, x.r, x.r² respectively  ...(2)

PT : x^b–c, y^c–a, z^a–b = 1

Substituting (1) and (2) in L.H.S., we get

L.H.S. = `x^("a" + "d" - "a" - 2"d") xx (x"r")^("a" + 2"d" - "a") xx (x"r"^2)^("a" - "a" - "d")`

= (x)^–d . (xr)^2d (xr²)^–d 

= `1/x^4 xx x^(2"d") *  "r"^(2"d") xx 1/(x^"d" "r"^(2"d")` = 1 = R.H.S.

Hence proved.