If a and b are coefficients of


If $a$ and $b$ are coefficients of $x^{n}$ in the expansions of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ respectively, then write the relation between a and $b$.


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

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

Thus we have.

${ }^{2 n} C_{n}=\frac{2 n !}{n ! n !}=\frac{2 n(2 n-1) !}{n(n-1) ! n !} \quad \ldots(1)$

and ${ }^{2 n-1} C_{n}=\frac{(2 n-1) !}{n !(n-1) !}$$\ldots(2)$

Dividing equation (1) by $(2)$, we get

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

$\Rightarrow \frac{a}{b}=2$

$\Rightarrow a=2 b$

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