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NCERT Class XI Mathematics - Binomial Theorem - Solutions

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Question : 25
Total: 36
Prove that the coefficient of xn in the expansion of (1+x)2n is twice thecoefficient of xn in the expansion of (1+x)2n–1.
Solution:  
Suppose xn occurs in the (r + 1)th term of the expansion (1+x)2n.
Now Tr+1 =
2n
‌
Crxr
On comparing power of x in xn and Tr+1, we get r = n
Thus the coefficient of xn is
2n
‌
Cn =
2n!
n!n!
=
2n(2n−1)!
n!n(n−1)!
 =
2(2n−1)!
n!(n−1)!
 ... (i)
Now suppose xn occurs in the (r + 1)th term of the expansion (1+x)2n−1
Now Tr+1 =
2n−1
‌
Crxr
On comparing power of x in xn and Tr+1, we get r = n
Thus the coefficients of xn =
2n−1
‌
Cn =
(2n−1)!
n!(n−1)!
 ... (ii)
From (i) & (ii), we see that coefficient of xn in the expansion of (1+x)2n is twice the coefficient of xn in the expansion of (1+x)2n–1.
Hence proved.
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