Question

How to prove ?

If `a/b = c/d = e/f`
prove that: `1/27 ((a+b)/b + (c+d)/d + (e+f)/f)^3 = [(a+b)(c+d)(e+f)]/(bdf)`

Answer 1

We start by expanding the LHS of the equation.

`1/27 ( a/b + c/d + e/f + 3)^3`

Let `x = a/b = c/d = e/f`. Then our LHS is `1/27(3x+3)^3`.

This can be further simplified: `1/27 (3x+3)^3 = 1/27 (3*(x+1))^3 = 27/ 27 (x+1)^3 = (x+1)^3`

We move to the RHS. Notice that

`((a+b)(c+d)(e+f))/(bdf) = (a+b)/b * (c+d)/d * (e+f)/f`
`= (a/b + 1)(c/d + 1)(e/f + 1) = (x+1)^3`.

Indeed, both the LHS and RHS simplify to the same expression, so they must be equivalent.