> 
> Calculate the sum of n elements of the sequence $a\_n$, in which:
>  $a\_1=3$, $a\_2=33$, $a\_3=333$, $a\_4=3333$ and so forth.
> 
> 
> 

We see that it's not an arithmetic progression as 3333-333 is not equal to 333-33 and so on.
It also isn't geometric progression as 3333/333 is not equal to 333/33 and so on.

Though, we see the corelation of: $a\_2=10\*a\_1+3$, $a\_3=10\*a\_2+3$ and so on. Then, the sum is: $10^0\*3+(10^1\*3+3)+(10^2\*3+10^1\*3+3)+...+(10^{n-1}\*3+10^{n-2}\*3+...+10^0\*3)$ - but is there any way to shorten it? I can't see any.