Class 11

Math

Algebra

Sequences and Series

The AM between two positive numbers a and $b(a>b)$ is twice their GM. Prove that $a:b=(2+3 ):(2−3 )$.

Given, $AM=2(GM)$

$⇒21 (a+b)=2ab ⇒2ab a+b =12 $

$⇒a+b−2ab a+b+2ab =2−12+1 $

$⇒(a −b )_{2}(a +b )_{2} =(1)_{2}(3 )_{2} $

$⇒a −b a +b =13 $

$⇒(a +b )−(a −b )(a +b )+(a −b ) =3 −13 +1 $

$⇒b a =3 −13 +1 $

$⇒ba =(3 −1)_{2}(3 +1)_{2} =2−3 2+3 $.