1．（a）By considering the series $$1 + t + t ^ { 2 } + t ^ { 3 } + \ldots + t ^ { n }$$ or otherwise，sum the series $$1 + 2 t + 3 t ^ { 2 } + 4 t ^ { 3 } + \ldots + n t ^ { n - 1 }$$ for \(t \neq 1\) ．
（b）Hence find and simplify an expression for $$1 + 2 \times 3 + 3 \times 3 ^ { 2 } + 4 \times 3 ^ { 3 } + \ldots + 2001 \times 3 ^ { 2000 }$$ （c）Write down an expression for both the sums of the series in part（a）for the case where \(t = 1\) ．