Question

Să se aducă la o forma mai simplă expresia:

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

Mulțumesc!​

Answer 1

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

2n+2n-1+...+n+1

n+1=2n-(n-1)

2n+2n-1+2n-2+...+2n-(n-1)

2n apare de n ori

Deci vom avea

2n×n-(1+2+3+...+n-1)=

$2n^2-\frac{(n-1)n}{2} =\frac{4n^2-n^2+n}{2} =\frac{3n^2+n}{2} =\frac{n(3n+1)}{2}$