$\binom{2n}{n+1}+\binom{2n}{n}$

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

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

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

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

Multiply $2(n+1)$ in numerator and denominator

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

$= \frac{\binom{2n+2}{n+1}}{2}$