The ratio of the coefficient of the middle term in the expansion of ( 1 + x ) 20 and the sum of the coefficients of two middle terms in expansion of ( 1 + x ) 19 is

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Aicharya · Accepted Answer

In $(1+x)^n$, if $n$ is even, then the middle term is the $\frac{n}{2} + 1^{th}$ term. So, the coefficient of the middle term in $(1+x)^{20}$ is $C_{10}^{20}$. In $(1+x)^n$, if $n$ is odd, then the middle term is the $\frac{n+1}{2}^{th}$ and $\frac{n+3}{2}^{th}$ terms. So, the sum of the coefficients of the two middle terms in $(1+x)^{19}$ is $C_{9}^{19} + C_{10}^{19}$. So the required ratio is $\frac{C_{10}^{20}}{C_{9}^{19} + C_{10}^{19}} = \frac{C_{10}^{20}}{C_{10}^{20}} = 1$.

The ratio of the coefficient of the middle term in the expansion of ( 1 + x ) 20 and the sum of the…

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