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Question

The coefficient of xn in the expansion of fraction numerator 1 over denominator open parentheses 1 minus x close parentheses open parentheses 3 minus x close parentheses end fractionis

  1. fraction numerator 3 to the power of n plus 1 end exponent minus 1 over denominator 2.3 to the power of n plus 1 end exponent end fraction    
  2. fraction numerator 3 to the power of n plus 1 end exponent minus 1 over denominator 3 to the power of n plus 1 end exponent end fraction    
  3. 2 open parentheses fraction numerator 3 to the power of n plus 1 end exponent minus 1 over denominator 3 to the power of n plus 1 end exponent end fraction close parentheses    
  4. none of these.    

The correct answer is: fraction numerator 3 to the power of n plus 1 end exponent minus 1 over denominator 2.3 to the power of n plus 1 end exponent end fraction


    fraction numerator 1 over denominator left parenthesis 1 minus x right parenthesis left parenthesis 3 minus x right parenthesis end fraction equals fraction numerator 1 over denominator 2 end fraction open square brackets fraction numerator 1 over denominator 1 minus x end fraction minus fraction numerator 1 over denominator 3 minus x end fraction close square brackets = fraction numerator 1 over denominator 2 end fraction [(1 - x)-1 - (3 - x)-1]
    = fraction numerator 1 over denominator 2 end fraction open square brackets left parenthesis 1 minus x right parenthesis to the power of negative 1 end exponent minus 3 to the power of negative 1 end exponent open parentheses 1 minus fraction numerator x over denominator 3 end fraction close parentheses to the power of negative 1 end exponent close square brackets
    = fraction numerator 1 over denominator 2 end fraction open square brackets 1 plus x plus x to the power of 2 end exponent plus x to the power of 3 end exponent plus... negative fraction numerator 1 over denominator 3 end fraction open parentheses 1 plus fraction numerator x over denominator 3 end fraction plus open parentheses fraction numerator x over denominator 3 end fraction close parentheses to the power of 2 end exponent plus... close parentheses close square brackets
    coefficient of xn = fraction numerator 1 over denominator 2 end fraction open square brackets 1 minus fraction numerator 1 over denominator 3 end fraction times fraction numerator 1 over denominator 3 to the power of n end exponent end fraction close square brackets = fraction numerator 3 to the power of n plus 1 end exponent minus 1 over denominator 2 times 3 to the power of n plus 1 end exponent end fraction

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