The question is attached above.

Asked by sudhanshubhushanroy | 19th Nov, 2017, 09:07: PM

Expert Answer:

begin mathsize 16px style straight a subscript straight n equals begin inline style sum from straight r equals 0 to straight n of end style fraction numerator 1 over denominator straight C presuperscript straight n subscript straight r end fraction
Using space straight C presuperscript straight n subscript straight r equals straight C presuperscript straight n subscript straight n minus straight r end subscript space and space straight C presuperscript straight n subscript straight n equals 1 comma space straight C presuperscript straight n subscript 0 equals 1
you space get space
straight a subscript straight n equals 2 plus 2 open parentheses fraction numerator 1 over denominator straight C presuperscript straight n subscript 1 end fraction plus fraction numerator 1 over denominator straight C presuperscript straight n subscript 2 end fraction plus....... close parentheses
rightwards double arrow fraction numerator straight a subscript straight n minus 2 over denominator 2 end fraction equals fraction numerator 1 over denominator straight C presuperscript straight n subscript 1 end fraction plus fraction numerator 1 over denominator straight C presuperscript straight n subscript 2 end fraction plus............ left parenthesis 1 right parenthesis
To space find space
begin inline style sum from straight r equals 0 to straight n of end style begin inline style space end style begin inline style fraction numerator straight r over denominator straight C presuperscript straight n subscript straight r end fraction end style begin inline style equals end style begin inline style 0 end style begin inline style plus end style begin inline style fraction numerator 1 over denominator straight C presuperscript straight n subscript 1 end fraction end style begin inline style plus end style begin inline style fraction numerator 2 over denominator straight C presuperscript straight n subscript 2 end fraction end style begin inline style plus end style fraction numerator 3 over denominator straight C presuperscript straight n subscript 3 end fraction begin inline style plus end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style plus end style begin inline style fraction numerator straight n minus 2 over denominator straight C presuperscript straight n subscript straight n minus 2 end subscript end fraction end style begin inline style plus end style fraction numerator straight n minus 1 over denominator straight C presuperscript straight n subscript straight n minus 1 end subscript end fraction begin inline style plus end style fraction numerator straight n over denominator straight C presuperscript straight n subscript straight n end fraction
begin inline style equals end style begin inline style open parentheses fraction numerator 1 over denominator straight C presuperscript straight n subscript 1 end fraction plus fraction numerator straight n minus 1 over denominator straight C presuperscript straight n subscript straight n minus 1 end subscript end fraction close parentheses end style begin inline style plus end style begin inline style open parentheses fraction numerator 2 over denominator straight C presuperscript straight n subscript 2 end fraction plus fraction numerator straight n minus 2 over denominator straight C presuperscript straight n subscript straight n minus 2 end subscript end fraction close parentheses end style begin inline style plus end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style space end style begin inline style equals end style begin inline style open parentheses fraction numerator 1 over denominator straight C presuperscript straight n subscript 1 end fraction plus fraction numerator straight n minus 1 over denominator straight C presuperscript straight n subscript 1 end fraction close parentheses end style begin inline style plus end style begin inline style open parentheses fraction numerator 2 over denominator straight C presuperscript straight n subscript 2 end fraction plus fraction numerator straight n minus 2 over denominator straight C presuperscript straight n subscript 2 end fraction close parentheses end style begin inline style plus end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style space end style begin inline style using end style begin inline style space end style begin inline style blank presuperscript straight n end style begin inline style straight C subscript straight r end style begin inline style equals end style begin inline style blank presuperscript straight n end style begin inline style straight C subscript straight n minus straight r end subscript end style
equals begin inline style fraction numerator straight n over denominator straight C presuperscript straight n subscript 1 end fraction end style begin inline style plus end style begin inline style fraction numerator straight n over denominator straight C presuperscript straight n subscript 2 end fraction end style begin inline style plus end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style. end style begin inline style plus end style begin inline style fraction numerator straight n over denominator straight C presuperscript straight n subscript straight n end fraction end style plus straight n over 1
equals straight n open parentheses fraction numerator straight a subscript straight n minus 2 over denominator 2 end fraction plus 1 close parentheses............... from space left parenthesis 1 right parenthesis
equals na subscript straight n over 2 end style

Answered by Sneha shidid | 30th Nov, 2017, 10:27: AM