Challenging +1.8 This AEA question requires differentiating a geometric series to find a weighted sum, then applying the result with substitution. While it involves multiple sophisticated steps (algebraic manipulation of series, differentiation technique, substitution and simplification), the path is relatively guided through parts (a)-(c). The techniques are advanced but systematic rather than requiring deep insight, placing it above typical A-level but below the most challenging AEA problems.
5.(a)Show that
$$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$
(b)Hence find an expression in terms of \(x\) and \(n\) for \(\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }\) for \(x \neq 0\) and \(x \neq 1\)
Simplify your answer.
(c)Find \(\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)\)
Give your answer in the form \(a - \frac { b + c n } { 2 ^ { n } }\) ,where \(a , b\) and \(c\) are integers.
5.(a)Show that
$$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$
(b)Hence find an expression in terms of $x$ and $n$ for $\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }$ for $x \neq 0$ and $x \neq 1$\\
Simplify your answer.\\
(c)Find $\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)$
Give your answer in the form $a - \frac { b + c n } { 2 ^ { n } }$ ,where $a , b$ and $c$ are integers.\\
\hfill \mbox{\textit{Edexcel AEA 2016 Q5 [13]}}