Question

If a, b, c, d are in continued proportion, prove that: `(a^3 + b^3 + c^3)/(b^3 + c^3 + d^3) = a/d`

Answer 1

a, b, c, d are in continued proportion

∴ `a/b = b/c = c/d` = k(say)

∴ c = dk, b = ck = dk², a = bk = dk³ 

L.H.S.

= `(a^3 + b^3 + c^3)/(b^3 + c^3 + d^3)`

= `((dk^3)^3 + (dk^2)^3 + (dk)^3)/((dk^2)^3 + (dk)^3 + d^3)`

= `(d^3k^9 + d^3k^6 + d^3k^3)/(d^3k^6 + d^3k^3 + d^3)`

= `(d^3k^3(k^6 + k^3 + 1))/(d^3(k^6 + k^3 + 1)`

= k³

R.H.S.

= `a/d`

= `(dk^3)/d`

= k³

∴ L.H.S. = R.H.S.