Combinatorial proof of $1+2(\sum_{i=0}^n 3^i)=3^{n+1}$ I have this workbook of proofs that I've been trying to finish for a couple of months now. There is this problem in it that requires me to prove $1+2(\sum_{i=0}^n 3^i)=3^{n+1}$ using combinatorial identities only. This problem has stumped me for several days and I would appreciate any help I could get

Here's a different solution than in the link:
consider a tournament with $3^{n+1}$ players; divide them into groups of three, with two games per group(every game an elimination), and let the winner proceed to the next round, until we have a single final winner from the last single group of three. 
The number of games played is the LHS (excluding the -1 term), but also because every game eliminates one player, it is $3^{n+1}-1$.
