Question

If a, b, c are in continued proportion and a(b – c) = 2b, prove that: `a - c = (2(a + b))/a`.

Answer 1

Since a, b, c are in continued proportion,

`a/b = b/c`

`\implies` b² = ac  ...(i)

a(b – c) = 2b

`\implies (a(b - c))/b = 2`  ...(ii)

Now,

R.H.S. = `(2(a + b))/a`

`\implies (a(b - c))/b xx (a + b)/a`   ...Using equation (ii)

`\implies (cancela(b - c))/b xx (a + b)/cancela` 

`\implies (b - c)/b xx a + b`

`\implies (ba + b^2 - ac - bc)/b`

`\implies (ba + ac - ac - bc)/b`    ...(Using equation (I))

`\implies (ba - bc)/b`

`\implies (cancel(b)(a - c))/cancel(b)`

`\implies` a – c