Relations and Functions

Relations and Functions

Question

Simplify

space space space space space space space space space space space space space space space space fraction numerator straight i to the power of 4 straight n space plus space 1 end exponent minus straight i to the power of 4 straight n minus 1 end exponent over denominator 2 end fraction

Answer
space space space space space space space space space space space space space space space space space fraction numerator straight i to the power of 4 straight n plus 1 end exponent minus straight i to the power of 4 straight n minus 1 end exponent over denominator 2 end fraction equals fraction numerator straight i to the power of 4 straight n end exponent. straight i to the power of 1 minus straight i to the power of 4 straight n end exponent. straight i to the power of negative 1 end exponent over denominator 2 end fraction equals fraction numerator straight i minus straight i to the power of negative 1 end exponent 2 over denominator 2 end fraction                   (   i4n = 1)
  
    space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator straight i minus begin display style 1 over straight i end style over denominator 2 end fraction equals fraction numerator straight i minus begin display style 1 over straight i end style. begin display style straight i cubed over straight i cubed end style over denominator 2 end fraction equals fraction numerator straight i minus straight i cubed over denominator 2 end fraction

space space space space space space space space space space space space space space space space space space space space space space space space                             (  i4n = 1)
             space space space space space space space space space space equals space fraction numerator straight i minus left parenthesis negative straight i right parenthesis over denominator 2 end fraction                                                 
             space space space space space space space space space space equals space fraction numerator 2 straight i over denominator 2 end fraction equals straight i

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