AM and GM (Arithmetic Mean And Geometric mean)
AM and GM (Arithmetic Mean And Geometric mean)
Last updated at April 16, 2024 by Teachoo
Misc 10 The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that a : b = (m + ( ^2 ^2 )) : (m ( ^2 ^2 ) ) Introduction Componendo dividendo If / = / Applying componendo dividendo ( + )/( ) = ( + )/( ) Eg: Taking 1/2 = 4/8 (1+ 2)/(1 2) = (4 + 8)/(4 8) 3/( 1) = 12/( 4) -3 = -3 Misc 19 The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that a : b = (m + ( ^2 ^2 )) : (m ( ^2 ^2 ) ) Here, the two numbers be a and b. Arithmetic Mean =AM= (a+b)/2 & Geometric Mean=GM= ab According to the question, AM/( GM" " ) = / ( + )/(2 " " ) = / Applying componendo dividendo ( + +2 )/( + 2 ) = ( + )/( ) (( )2+( )2+2( ))/(( )2+( )2 2( ) ) =( + )/( ) Using (x + y)2 = x2 + y2 + 2xy (x - y)2 = x2 + y2 - 2xy ( + )2/( )2 = ( + )/( ) (( + )/( ))^2 = ( + )/( ) ( + )/( ) = (( + )/( )) ( + )/( ) = ( + )/( ( ) ) Applying componendo dividendo (( + ) + ( ))/(( + ) ( ) ) = ( ( + ) + ( ))/( ( + ) ( )) (2 )/(2 ) = ( ( + ) + ( ))/( ( + ) ( )) / = ( ( + ) + ( ))/( ( + ) ( )) Squaring both sides ( / )^2 = (( ( + ) + ( ))/( ( + ) ( )))^2 ( )^2/( )^2 = ( ( + ) + ( ))^2/( ( + ) ( ))^2 Using (x + y)2 = x2 + y2 + 2xy (x - y)2 = x2 + y2 - 2xy / = (( ( + ) )^2+( ( ) )^2+ 2( ( + ))( ( )))/(( ( + ) )^2+( ( ) )^2 2( ( + ))( ( )) ) / = ( + + + 2 (( + )( ) ))/( + + 2 (( + )( ) )) / = ( + + + 2 (( ^2 ^2 ) ))/( + + 2 (( ^2 ^2 ) )) / = (2 + 2 (( ^2 ^2 ) ))/(2 2 (( ^2 ^2 ) )) / = 2( + (( ^2 ^2 ) ))/2( (( ^2 ^2 ) )) / = ( + (( ^2 ^2 ) ))/( (( ^2 ^2 ) )) Thus, a : b = (m + ( ^2 ^2 )) : (m ( ^2 ^2 ) ) Hence proved