The sum to $$n$$ terms of the series $$2 + 5 +14 + 41 + . . . . .\,$$     is

Solution :
$$\eqalign{ & {\text{Let, }}{S_n} = 2 + 5 + 14 + 41 + ..... + {x_n} \cr & {S_n} = 2 + 5 + 14 + ..... + {x_{n - 1}} + {x_n} \cr & 0 = 2 + \left[ {3 + 9 + 27 + .....\,{\text{to}}\left( {n - 1} \right)\,{\text{terms}}} \right] - {x_n} \cr & \therefore {x_n} = 2 + \frac{{3\left( {{3^{n - 1}} - 1} \right)}}{{3 - 1}} = \frac{1}{2} + \frac{1}{2} \cdot {3^n} \cr & \therefore {S_n} = \sum {{x_n}} = \frac{1}{2}\sum 1 + \frac{1}{2}\sum {{3^n}} \cr & = \frac{n}{2} + \frac{1}{2} \cdot \frac{{3\left( {{3^{n - 1}} - 1} \right)}}{{3 - 1}} = \frac{n}{2} + \frac{3}{4}\left( {{3^n} - 1} \right) \cr} $$