If $a$and $b$are the greatest values of ${ }^{2n}{C}_r{\text{ and }}^{2n-1}{C}_r$ respectively. Then 

We know that the greatest value of ${ }^n{C}_r$ is given by

$\left\{\begin{array}{l}{ }^n{C}_{n/2}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }n\text{ is even }\\ \frac{{ }^n{C}_{n-1}}{2}\text{ or }\frac{{ }^n{C}_{n+1}}{2},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }n\text{ is odd }\end{array}\right.$

$\therefore a=$greatest value of ${ }^{2n}{C}_r{=}^{2n}{C}_n$ 

and, 

$b=$ Greatest value of ${ }^{2n-1}{C}_r{=}^{2n-1}{C}_{n-1}$

$\Rightarrow \frac{a}{b}=\frac{{ }^{2n}{C}_n}{{ }^{2n-1}{C}_{n-1}}=\frac{{\frac{2n}{n}}^{2n-1}{C}_{n-1}}{2n-1_{C_{n-1}}}=2\Rightarrow a=2b$