If... #color(white)x^(x + 1)C_3 - ^(x - 1)C_3 = 16x = ?#

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Answer 1 · 2017-11-22T08:58:44

#x=5#

#C_3^(x+1)=((x+1)x(x-1))/(1.2.3)#

and #C_3^(x-1)=((x-1)(x-2)(x-3))/(1.2.3)#

Therefore #C_3^(x+1)-C_3^(x-1)#

= #((x+1)x(x-1))/(1.2.3)-((x-1)(x-2)(x-3))/(1.2.3)#

= #(x-1)/6{x(x+1)-(x-2)(x-3)}#

= #(x-1)/6{x^2+x-(x^2-5x+6)}#

= #(x-1)/6(6x-6)#

= #(x-1)^2#

Hence #C_3^(x+1)-C_3^(x-1)=16x# is equivalent to

#(x-1)^2=16#

or #x-1=+-4#

i.e. #x=5# or #-3#

and if we consider only as a positive integer #x=5#