Question

If a, b, c, d are in continued proportion, prove that: a : d = triplicate ratio of (a – b) : (b – c)

Answer 1

a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = ck = dk. k = dk²,
a = bk = dk². k = dk³ 
a : d = triplicate ratio of (a – b) : (b – C)
= (a – b)³ : (b – c)³ 
L.H.S. = a  : d 
= `a/d`

= `(dk^3)/d`
= k³
R.H.S. = `(a - b)^3/(b - c)^3`

= `(dk^3 - dk^2)^3/(dk^2 - dk)^3`

= `(d^3k^6(k - 1)^3)/(d^3k^3(k - 1)^3`
= k³
∴ L.H.S. = R.H.S.