Permutations and Combinations

Permutations and Combinations

Question

Find the sum of n terms of sequence

open parentheses straight x plus 1 over straight x close parentheses squared comma space space open parentheses straight x squared plus 1 over straight x squared close parentheses squared comma space space open parentheses straight x cubed plus 1 over straight x cubed close parentheses squared comma space space............

Answer

Let space space straight S subscript straight n denote the sum of first n terms of the given sequence.

∴            straight S subscript straight n space equals space open parentheses straight x plus 1 over straight x close parentheses squared plus open parentheses straight x squared plus 1 over straight x squared close parentheses squared plus open parentheses straight x plus 1 over straight x cubed close parentheses squared plus......... plus open parentheses straight x to the power of straight n plus 1 over straight x to the power of straight n close parentheses squared
                 equals open parentheses straight x squared plus 1 over straight x squared plus 2 close parentheses space plus space open parentheses straight x to the power of 4 plus 1 over straight x to the power of 4 plus 2 close parentheses plus open parentheses straight x to the power of 6 plus 1 over straight x to the power of 6 plus 2 close parentheses plus..... plus open parentheses straight x to the power of 2 straight n end exponent plus 1 over straight x to the power of 2 straight n end exponent plus 2 close parentheses
                 = left square bracket straight x squared plus straight x to the power of 4 plus straight x to the power of 6 plus......... straight x to the power of 2 straight n end exponent right square bracket space plus space open square brackets 1 over straight x squared plus 1 over straight x to the power of 4 plus 1 over straight x to the power of 6 plus...... plus 1 over straight x to the power of 2 straight n end exponent close square brackets plus 2 straight n
                = fraction numerator straight x squared left square bracket left parenthesis straight x squared right parenthesis to the power of straight n minus 1 right square bracket over denominator straight x squared minus 1 end fraction plus fraction numerator begin display style 1 over straight x squared end style open square brackets 1 minus open parentheses begin display style 1 over straight x squared end style close parentheses to the power of straight n close square brackets over denominator 1 minus begin display style 1 over straight x squared end style end fraction plus 2 straight n space equals space straight x squared open parentheses fraction numerator straight x to the power of 2 straight n minus 1 end exponent over denominator straight x squared minus 1 end fraction close parentheses plus fraction numerator left parenthesis straight x to the power of 2 straight n end exponent minus 1 right parenthesis over denominator straight x to the power of 2 straight n end exponent left parenthesis straight x squared minus 1 right parenthesis end fraction plus 2 straight n
                                                                         open curly brackets because space space In space straight G. straight P. comma space space straight S subscript straight n space equals space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction close curly brackets
               equals open parentheses fraction numerator straight x to the power of 2 straight n end exponent minus 1 over denominator straight x squared minus 1 end fraction close parentheses open parentheses straight x squared plus 1 over straight x to the power of 2 straight n end exponent close parentheses plus 2 straight n

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