| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2013 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Sum of Powers Using Standard Formulae |
| Difficulty | Challenging +1.2 This is a structured multi-part question that guides students through each step systematically. Parts (a)-(c) are routine (writing terms, verifying a recurrence, summing a geometric series). Parts (d)-(f) require inequality manipulation and geometric series bounds, but the scaffolding makes the path clear. While it's an AEA question, the extensive guidance and standard techniques place it only moderately above average difficulty. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
4.A sequence of positive integers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ has $r$ th term given by
$$a _ { r } = 2 ^ { r } - 1$$
\begin{enumerate}[label=(\alph*)]
\item Write down the first 6 terms of this sequence.
\item Verify that $a _ { r + 1 } = 2 a _ { r } + 1$
\item Find $\sum _ { r = 1 } ^ { n } a _ { r }$
\item Show that $\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }$
\item Hence show that $1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)$
\item Show that $\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2013 Q4 [13]}}