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Question

If text end text stretchy sum from r equals 0 to n of   open curly brackets a subscript r end subscript left parenthesis x minus y plus 2 right parenthesis to the power of r end exponent minus b subscript r end subscript left parenthesis y minus x minus 1 right parenthesis to the power of r end exponent close curly brackets equals 0 for all n element of N comma text  then  end text b subscript n end subscript text end textis equal to

  1. a subscript n end subscript    
  2. left parenthesis negative 1 right parenthesis to the power of n end exponent a subscript n end subscript    
  3. left parenthesis negative 1 right parenthesis to the power of n end exponent 2 to the power of n end exponent a subscript n end subscript    
  4. 2 to the power of n end exponent a subscript n end subscript    

The correct answer is: left parenthesis negative 1 right parenthesis to the power of n end exponent a subscript n end subscript


    text Take  end text y minus x minus 1 equals X rightwards double arrow stretchy sum from r equals 0 to n of   a subscript r end subscript left parenthesis 1 minus X right parenthesis to the power of r end exponent equals stretchy sum from r equals 0 to n of   b subscript r end subscript X to the power of r end exponent
    text Equating coefficient of  end text X to the power of n end exponent text  both sides  end text rightwards double arrow b subscript n end subscript equals left parenthesis negative 1 right parenthesis to the power of n end exponent a subscript n end subscript.

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