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Question

The sum of coefficients of the two middle terms in the expansion of (1+x)2n1 is equal to 2n1Cn. State true or false.


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Solution

The general term of the expansion (a+b)n is given by

Tr+1=nCranrbr

Given: (1+x)2n1

Tr+1=2n1Cr(1)2n1r(x)r=2n1Cr(x)r

If n is odd, then the middle terms are (n+12)thand (n+32)th term

(2n1+1)2=n

(2n1+3)2=n+1

So, the middle terms are nth term and (n+1)th term

Coefficient of n^th term in the expansion = 2n1Cn1

Coefficient of (n+1)th term in the expansion = 2n1Cn

Therefore, the sum of coefficients of the two middle terms

=2n1Cn1+2n1Cn

=2nCn

[nCr+nCr1=n+1Cr]

Hence, the given statement is False.


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