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Misc 19 - Ratio of AM and GM of a, b is m : n. Show that - AM and GM (Arithmetic Mean And Geometric mean)

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Misc 19 - Chapter 9 Class 11 Sequences and Series - Part 2
Misc 19 - Chapter 9 Class 11 Sequences and Series - Part 3 Misc 19 - Chapter 9 Class 11 Sequences and Series - Part 4 Misc 19 - Chapter 9 Class 11 Sequences and Series - Part 5

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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 ) ) 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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.