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NDA Math Binomial Theorem

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Consider the expansion of

(1+x)^{2n+1}

[NDA II 2015]

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Question : 39 of 50

Marks: +1, -0

The average of the coefficients of the two middle terms in the expansion is

${}^{2n+1}C_{n+2}$
${}^{2n+1}C_n$
${}^{2n+1}C_{n-1}$
${}^{2n}C_{n+1}$

Solution:

Total no. of terms in the expansion is 2n + 2. The middle two terms will be nth, (n + l)th term. So.

Average

= \frac{(2n+1)c_{n}+(2n+1)c_{n+1}}{2}

= \frac{ \left[ \frac{(2n+1)!}{n!(n+1)!} + \frac{(2n+1)!}{(n+1)!n!} \right] }{2}

= \frac{(2n+1)!}{n!(n+1)!}

= (2n+1)C_{n}

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