If $a$ is the coefficient of middle term in the expansion of ${\left(1+x\right)}^{2n}$ and $b,c$are the coefficients of two middle terms in the expansion of ${\left(1+x\right)}^{2n-1}$ then 

Given $a$ is coefficient of middle term in the expansion of ${\left(1+x\right)}^{2n}$ then $a=2n_{C_{n+1}}$

Given $b,c$ are coefficients of middle terms in the expansion of ${\left(1+x\right)}^{2n-1}$ then $b={\left(2n-1\right)}_{C_n}$ and $c={\left(2n-1\right)}_{C_{n+1}}$

We know that $n_{C_r}+n_{C_{r-1}}=n+1_{C_r}$

Hence, ${\left(2n-1\right)}_{C_{n+1}}+{\left(2n-1\right)}_{C_n}={\left(2n\right)}_{C_{n+1}}$

Therefore, $\boxed{b+c=a}$