Standard +0.8 This AEA question requires differentiating a geometric series to obtain a new series sum (a non-standard technique), then applying it with specific values and handling the special case t=1. While methodical once the approach is seen, it demands mathematical maturity beyond typical A-level, involving series manipulation and careful algebraic simplification across multiple parts.
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 \(t = 1\) .
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 $t = 1$ .
\hfill \mbox{\textit{Edexcel AEA 2002 Q1 [7]}}