Question

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

Answer 1

answer solved in the attachment

Answer 2

Answer:

a , b , c , d are in GP

Step-by-step explanation:

(a + bx)/(a - bx)  = (b + cx)/(b - cx) = (c + dx)/(c - dx)

Taking 1st two

(a + bx)/(a - bx)  = (b + cx)/(b - cx)

=> (a + bx)(b - cx) = (a - bx)(b + cx)

=> ab - bcx² - acx + b²x = ab  + acx  - b²x -bcx²

Canceeling ab - bcx²

=> - acx + b²x =  acx  - b²x

dividing by x both sides

=> -ac + b² = ac - b²

=> 2ac = 2b²

=> ac = b²

=> a , b , c are in GP

Taking Last two

(b + cx)/(b - cx)  =  (c + dx)/(c - dx)

=> (b + cx)(c - dx) = (b - cx)(c + dx)

=> bc - bdx + c²x - cdx² = bc + bdx - c²x -cdx²

cancelling bc  - cdx²

=> c²x  - bdx = bdx - c²x

=> 2c²x  = 2 bdx

=> c² = bd

=> b , c , d are in GP

a , b ,c , in GP   &  b  , c , d  in GP

=> a , b , c , d are in GP