Prove that the greatest coefficient in the expansion of $(1 + x)^{2 n}$ is double the greatest coefficient in the expansion $(1 + x)^{2 n - 1}$ .

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Solution

greatest coefficient of $(1 + x)^{n}$ is given by
(i) r = n/2 when n is even
(ii) r = (n-1)/2 or (n+1)/2 when n is odd
Greatest coefficient of $(1 + x)^{2 n}$ is $^{2 n} C_{2 n / 2} =^{2 n} C_{n}$
Greatest coefficient of $(1 + x)^{2 n - 1}$ is $^{2 n - 1} C_{2 n - 1 + 1 / 2}$ or $^{2 n - 1} / - 1 + 2$
Consider
$^{2 n - 1} C_{2 n - 1 + 1 / 2} =^{2 n - 1} C_{2 n / 2} =^{2 n - 1} C_{n}$
$= \frac{2 n - 1!}{n - 1! n!}$
$= \frac{2 n}{2 n} \frac{2 n - 1!}{n - 1! n!}$
$= \frac{2 n!}{2 n! n!}$
$= {\frac{1}{2}}^{2 n} C_{n}$
$= 1 / 2$ Greatest coff. of $(1 + x)^{2 n} . . . (1)$
$^{2 n - 1} C_{2 n + 1 / 2} =^{2 n - 1} C_{n - 1} = \frac{2 n - 1!}{n - 1! n!} = \frac{2 n}{2 n} \frac{2 n - 1!}{n - 1! n!}$
$= {\frac{1}{n}}^{2 n} C_{n} . . . (1)$
From (I) & (II)
Greatest coefficient of $(1 + x)^{2 n}$ is twice greatest coefficient
of $(1 + x)^{2 n - 1}$

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Prove that the coefficient of $x^{n}$ in the binomial expansion of $(1 + x)^{2 n}$ is twice the coefficient of $x^{n}$ in the binomial expansion of $(1 + x)^{2 n - 1}$ .

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Prove that the greatest coefficient in the expansion of (1+x)2n is double the greatest coefficient in the expansion (1+x)2n−1.

Prove that the greatest coefficient in the expansion of $(1 + x)^{2 n}$ is double the greatest coefficient in the expansion $(1 + x)^{2 n - 1}$ .