If a + bx / a-bx = b + cx / b - cx = c + dx/c - dx (x ) (x ≠ 0) then show that a, b, c, d are in GP.

Question

If \(\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\) (x \(\ne\) 0) then show that a, b, c, d are in GP.

Answer 1

 \(\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} \)  (Given data in the question) → (b)

Cross multiplying (1) and expanding, 

(a + bx)(b – cx) = (b + cx)(a-bx) 

ab – acx + b²x – bcx² = ba –b²x + acx – bcx² 

2b²x = 2acx b² = ac → (i) 

If three terms are in GP, then the middle term is the Geometric Mean of first term and last term. 

→ b² = ac So, 

from (i) b, is the geometric mean of a and b. 

So, a, b, c are in GP.

\(\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} \)  (Given data in the question)  → (2)

Cross multiplying (2) and expanding, 

(b + cx)(c – dx) = (c + dx)(b – cx) 

bc – bdx + c²x – cdx² = cb – c²x + bdx – dcx² 

2c²x = 2bdx 

c² = bd → (ii)

So, from (ii), c is the geometric mean of b and d. 

So, b, c, d is in GP. 

∴ a, b, c, d are in GP.