Question

If three quantities are in continued proporation show that the ratio of the first to the third is the duplicate ratio of the first to the second

Answer 1

let x ,y ,z ,be the three quantities which are in continued proportion

⇒x/y =y/z ..........,[1]

⇒y^2 =xz ..........[2]

squaring [1] we get

[x/y]^2 = [y/z]^2 ⇒ y^2/z^2 =x^2/y^2

⇒xz/y^2 =x^2 /y^2

⇒x/z =x^2/y^2

x : z =x^2 ; y^2

ie,the ratio of the first to the third is the duplicate ratio of the first to the second.

Answer 2

let x ,y ,z ,be the three quantities which are in continued proportion  

then  x : y : z ⇒y^2=xz ....[1]

Now ,we have to prove that  

⇒x : z =x^2 : y^2

That is we need to prove that

⇒  x y^2=x^2 z

⇒LHS =x y^2 =x [xz] =x^2 z =RHS  [using 1]

Hence proved