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Let n be a positive integer. Show that $\binom{2n}{n + 1} + \binom{2n}{n} = \dfrac{\binom{2n + 2}{n + 1}}{2}.$

simply after simplification we get it.
$\binom{2n}{n+1}+\binom{2n}{n}$

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

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

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

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

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

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

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