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.