Summing the RHS of all those equations we have:
1 + x + 2x + x2 + 2x2 + 2x2 + x3 + 2x3 + 2x3 + 2x3 ...
= 1 + x + x2 + x3 + 2x + 2x2 + 2x2 + 2x3 + 2x3 + 2x3 ...
= 1 + x + x2 + x3 + 2x + 4x2 + 6x3 ...
= sum_{n=0}^{\infty} xn + 2nxn
= sum_{n=0}^{\infty} (1+2n)xn