| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove summation with exponentials |
| Difficulty | Standard +0.3 This is a standard two-part induction question from Further Pure 1. Part (i) involves proving a recurrence relation formula using straightforward induction with simple algebraic manipulation. Part (ii) requires induction on a summation with exponentials, which is slightly more involved algebraically but follows a completely standard template. Both parts are routine exercises that test basic induction technique without requiring novel insight or complex problem-solving, making this slightly easier than average for A-level. |
| Spec | 4.01a Mathematical induction: construct proofs |
\begin{enumerate}
\item (i) A sequence of positive numbers is defined by
\end{enumerate}
$$\begin{aligned}
u _ { 1 } & = 5 \\
u _ { n + 1 } & = 3 u _ { n } + 2 , \quad n \geqslant 1
\end{aligned}$$
Prove by induction that, for $n \in \mathbb { Z } ^ { + }$,
$$u _ { n } = 2 \times ( 3 ) ^ { n } - 1$$
(ii) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$,
$$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$
\hfill \mbox{\textit{Edexcel F1 2018 Q10 [11]}}