Question

If $\frac{ a + bx }{ a - bx }= \frac{b + cx }{ b - cx} =\frac{ c + dx }{ c - dx }(x ≠ 0),$ then show that a, b, c and d are in G.P.

Answer 1

It is given that,
$\frac{ a + bx }{ a - bx }= \frac{b + cx }{ b - cx} =\frac{ c + dx }{ c - dx }$
$⇒(a + bx)(b - cx) = (b + cx)(a - bx)$
$⇒ab - acx + b^2x - bcx^2 = ab - b^2x + acx - bcx^2$
$⇒2b^2x = 2acx$
$⇒ b^2 = ac$
$⇒ \frac{b }{ a} =\frac{ c }{ b} ...(1)$
Also, $\frac{ b + cx }{ b - cx }= \frac{c + dx }{ c - dx}$
$⇒ (b + cx)(c - dx) = (b - cx)(c + dx)$
$⇒ bc - bdx + c^2x - cdx^2 = bc + bdx - c^2x - cdx^2$
$⇒ 2c^2x = 2bdx$
$⇒ c^2 = bd$
$⇒ \frac{c }{ d} = \frac{d }{c} ...(2)$
From (1) and (2), we obtain
$\frac{b }{ a} =\frac{ c }{ b} =\frac{ d }{ c}$
Thus, a, b, c, and d are in G.P.