Let n be a positive even integer. If the rati | vedclass

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(B) Given that $n$ is a positive even integer, let $n = 2m$. In the expansion of $(1+x)^{n}$, the largest coefficient is the middle term, which is $^n

Vedclass · Accepted Answer

$21$

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(B) Given that $n$ is a positive even integer, let $n = 2m$. In the expansion of $(1+x)^{n}$, the largest coefficient is the middle term, which is $^nC_{n/2} = ^{2m}C_m$. The $2^{nd}$ largest coefficients are the terms adjacent to the middle term, which are $^nC_{m-1}$ and $^nC_{m+1}$. Given the ratio $\frac{^nC_m}{^nC_{m-1}} = \frac{11}{10}$. Using the formula $\frac{^nC_r}{^nC_{r-1}} = \frac{n-r+1}{r}$, we have: $\frac{2m-m+1}{m} = \frac{11}{10}$ $\frac{m+1}{m} = \frac{11}{10}$ $10(m+1) = 11m$ $10m + 10 = 11m$ $m = 10$. Thus, $n = 2m = 2(10) = 20$. The number of terms in the expansion of $(1+x)^{n}$ is $n+1 = 20+1 = 21$.

Let $n$ be a positive even integer. If the ratio of the largest coefficient and the $2^{nd}$ largest coefficient in the expansion of $(1+x)^{n}$ is $11:10$, then the number of terms in the expansion of $(1+x)^{n}$ is:

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