General term $U\_r $ of a given series , Where $$\sum^{n}\_{r=1}U\_r=\frac{3n}{2n+1}$$

I can evaluate that $$ U\_1=\frac{3}{3}$$

$$ U\_2=\frac{1}{5}$$

$$ U\_3=\frac{3}{35}$$

$$ U\_4=\frac{1}{21}$$

$$ U\_5=\frac{1}{33}$$

How can I recognize a pattern ? 

Yes I can guess a pattern ! But how can I mathematically prove it ?