prove that the coefficient of the middle term in the expansion of (1+x)^2n is equal to the sum of the coefficient of the middle term in the expansion of (1+x)^2n-1

Asked by yashwantjain12 | 10th Jan, 2016, 05:22: PM

Expert Answer:

begin mathsize 14px style In space straight a space binomial space expansion comma If space straight n space is space even. space then space open parentheses straight n over 2 plus 1 close parentheses to the power of th space term space is space the space middle space term. If space straight n space is space odd comma space then space space open parentheses fraction numerator straight n plus 1 over denominator 2 end fraction close parentheses to the power of th space and space space open parentheses fraction numerator straight n plus 3 over denominator 2 end fraction close parentheses to the power of th space terms space are space two space middle space terms.  In space left parenthesis 1 plus straight x right parenthesis to the power of 2 straight n end exponent space comma space 2 straight n space is space even. therefore The space middle space term space is space open parentheses fraction numerator 2 straight n over denominator 2 end fraction plus 1 close parentheses to the power of th equals left parenthesis straight n plus 1 right parenthesis to the power of th term. straight T subscript straight n plus 1 end subscript equals straight C presuperscript 2 straight n end presuperscript subscript straight n space left parenthesis 1 right parenthesis to the power of 2 straight n minus left parenthesis straight n right parenthesis end exponent space straight x to the power of left parenthesis straight n right parenthesis end exponent equals straight C presuperscript 2 straight n end presuperscript subscript straight n space space straight x to the power of left parenthesis straight n right parenthesis end exponent Coefficient space of space middle space term space equals straight C presuperscript 2 straight n end presuperscript subscript straight n  In space left parenthesis 1 plus straight x right parenthesis to the power of 2 straight n minus 1 end exponent space comma space 2 straight n minus 1 space is space odd. therefore The space middle space terms space are space is space open parentheses fraction numerator 2 straight n minus 1 plus 1 over denominator 2 end fraction close parentheses to the power of th equals left parenthesis straight n right parenthesis to the power of th term space space and space space open parentheses fraction numerator 2 straight n minus 1 plus 3 over denominator 2 end fraction close parentheses to the power of th equals left parenthesis straight n plus 1 right parenthesis to the power of th term   straight T subscript straight n space end subscript equals space straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n minus 1 end subscript space left parenthesis 1 right parenthesis to the power of 2 straight n minus 1 minus left parenthesis straight n minus 1 right parenthesis end exponent space straight x to the power of left parenthesis straight n minus 1 right parenthesis end exponent equals straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n minus 1 end subscript space space straight x to the power of left parenthesis straight n minus 1 right parenthesis end exponent Coefficient space of space middle space term left parenthesis left parenthesis straight n right parenthesis to the power of th term right parenthesis space equals straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n minus 1 end subscript  straight T subscript straight n plus 1 end subscript equals straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n space left parenthesis 1 right parenthesis to the power of 2 straight n minus 1 minus left parenthesis straight n right parenthesis end exponent space straight x to the power of left parenthesis straight n right parenthesis end exponent equals straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n space space straight x to the power of left parenthesis straight n right parenthesis end exponent Coefficient space of space middle space term space left parenthesis left parenthesis straight n plus 1 right parenthesis to the power of th term right parenthesis equals straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n  Sum space of space the space coefficient space of space middle space term space in space the space expansion space of space left parenthesis 1 plus straight x right parenthesis to the power of 2 straight n minus 1 end exponent equals straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n minus 1 end subscript plus straight C presuperscript 2 straight n minus 1 end presuperscript subscript straight n equals straight C presuperscript left parenthesis 2 straight n minus 1 right parenthesis plus 1 end presuperscript subscript straight n equals straight C presuperscript 2 straight n end presuperscript subscript straight n Coefficient space of space middle space term space in space the space expansion space of space left parenthesis 1 plus straight x right parenthesis to the power of 2 straight n end exponent     end style

Answered by Vijaykumar Wani | 11th Jan, 2016, 12:09: PM