Question:
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 $b$.
Solution:
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
$\Rightarrow \frac{{ }^{2 n} C_{n}}{{ }^{2 n-1} C_{n}}=\frac{2 n(2 n-1) ! n !(n-1) !}{n(n-1) ! n !(2 n-1) !}$
$\Rightarrow \frac{a}{b}=2$
$\Rightarrow a=2 b$