Last updated at May 29, 2018 by Teachoo

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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

Miscellaneous

Misc 1

Misc 2

Misc 3 Important

Misc 4

Misc 5

Misc 6

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11

Misc 12

Misc 13

Misc 14

Misc 15

Misc 16 Important

Misc 17

Misc 18

Misc 19 Important You are here

Misc 20

Misc 21

Misc 22

Misc 23

Misc 24

Misc 25 Important

Misc 26

Misc 27

Misc 28 Important

Misc 29

Misc 30

Misc 31

Misc 32 Important

Chapter 9 Class 11 Sequences and Series

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.