The finite series 2+6+18+54+⋯+2⋅3n−1 (containing n terms) is given.
a. Part 1: (i) Express the series in summation notation with index k starting from 0.
b. Part 2: (ii) Write another valid summation expression for the same series with the index starting from k=1.
c. Part 3: (iii) Show that both representations obtained in parts (i) and (ii) yield identical numerical sums for every positive integer n, and briefly comment on why multiple summation forms are possible for the same series.