Q. 114.1( 55 Votes )
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n – 1.
Answer :
The general term Tr+1 in the binomial expansion is given by Tr+1 = nCr an-r br
The general term for binomial (1+x)2n is
Tr+1 = 2nCr xr …………………..1
To find the coefficient of xn
r=n
Tn+1 = 2nCn xn
The coefficient of xn = 2nCn
The general term for binomial (1+x)2n-1 is
Tr+1 = 2n-1Cr xr
To find the coefficient of xn
Putting n =r
Tr+1 = 2n-1Cr xn
The coefficient of xn = 2n-1Cn
We have to prove
Coefficient of xn in (1+x)2n = 2 coefficient of xn in (1+x)2n-1
Hence L.H.S = R.H.S.
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