Some summtion formulas like $\sum_{i=1}^{n}i = \frac{n(n+1)}{2}$ and,
$\sum_{i=1}^n\frac{i}{2^i} = 2- \frac{n+2}{2^n}$ hold and can be proven by induction.
However, I'm curious to know if there are any formulas that hold true only for the first few terms ($n \in [1,50]$), but fail to hold for larger values of $n$.
Thank you in advance