If a and b are the coefficients of

Question:

If $a$ and $b$ are the coefficients of $x^{n}$ in the expansion of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ respectively, find $\frac{a}{b}$.

Solution:

Coefficients of $x^{n}$ in the expansion of $(1+x)^{2 n}$ is ${ }^{2 n} C_{n}=a$.

Coefficients of $x^{n}$ in the expansion of $(1+x)^{2 n-1}$ is ${ }^{2 n-1} C_{n}=b$.

Now,

$\frac{a}{b}=\frac{{ }^{2 n} C_{n}}{{ }^{2 n-1} C_{n}}$

$=\frac{\frac{(2 n) !}{n ! n !}}{\frac{(2 n-1) !}{n !(n-1) !}}$

$=\frac{2 n}{n}$

$=2$

Hence, $\frac{a}{b}=2$.

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