Question

If `a/c = c/d = c/f` prove that : `bd f[(a + b)/b + (c + d)/d + (c + f)/f]^3` = 27(a + b)(c + d)(e + f)

Answer 1

`a/c = c/d = c/f` = k(say)
∴ a = bk, c = dk, e =fk
L.H.S. = `bd f[(a + b)/b + (c + d)/d + (c + f)/f]^3`

= `bd f[(bk + b)/b + (dk + d)/d + (fk + f)/f]^3`

= `bd f[(b(k + 1))/b + (d(k + 1))/d + (f(k + 1))/f]^3`

= bdf(k + 1 + k + 1 + k + 1)³
= bdf(3k + 3)³ = 27bdf(k + 1)³
R.H.S. = 27(a + b)(c + d)(e + f)
= 27(bk + b)(dk + d)(fk + f)
= 27b(k + 1)d(k + 1)f(k + 1)
= 27bdf(k + 1)³
∴ L.H.S. = R.H.S.