If a,b,c are in gp, a+h,b+h,c+h are in hp,then the value of h is?
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= ( 1/a + 1/b - 1/c )( 1/b + 1/c - 1/a )
= [ 1/b + ( 1/a - 1/c ) ] · [ 1/b - ( 1/a - 1/c ) ]
= (1/b)² - ( 1/a - 1/c )²
= 1/b² - [ ( 1/a² + 1/c² ) - 2/ac ]
= 1/b² - { [ ( 1/a + 1/c )² - 2/ac ] - 2/ac } ..... Note This Step
= 1/b² - [ ( 1/a + 1/c )² - 4/ac ]
= 1/b² - [ ( 2/b )² - 4/ac ] ................ from (1)
= 1/b² - 4/b² + 4/ac
= 4/ac - 3/b²
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