Q. 474.7( 3 Votes )

Prove that the co

Answer :

To Prove : coefficient of xn in (1+x)2n = 2 × coefficient of xn in (1+x)2n-1


For (1+x)2n,


a=1, b=x and m=2n


We have a formula,





To get the coefficient of xn, we must have,


xn = xr


• r = n


Therefore, the coefficient of xn


………



………..


………cancelling n


Therefore, the coefficient of xn in (1+x)2n………eq(1)


Now for (1+x)2n-1,


a=1, b=x and m=2n-1


We have formula,





To get the coefficient of xn, we must have,


xn = xr


• r = n


Therefore, the coefficient of xn in (1+x)2n-1




…..multiplying and dividing by 2


Therefore,


coefficient of xn in (1+x)2n-1 = � × coefficient of xn in (1+x)2n


or coefficient of xn in (1+x)2n = 2 × coefficient of xn in (1+x)2n-1


Hence proved.


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