| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove recurrence relation formula |
| 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 algebraic manipulation, while part (ii) is a divisibility proof requiring modular arithmetic. Both are textbook-style exercises with well-established techniques, making this slightly easier than average for Further Maths students but appropriately challenging for the syllabus. |
| Spec | 4.01a Mathematical induction: construct proofs |
8. (i) A sequence of numbers is defined by
$$\begin{aligned}
u _ { 1 } & = 3 \\
u _ { n + 1 } & = u _ { n } + 3 n - 2 \quad n \geqslant 1
\end{aligned}$$
Prove by induction that, for all positive integers $n$,
$$u _ { n } = \frac { 3 } { 2 } n ^ { 2 } - \frac { 7 } { 2 } n + 5$$
(ii) Prove by induction that, for all positive integers $n$,
$$f ( n ) = 3 ^ { 2 n + 3 } + 40 n - 27 \text { is divisible by } 64$$
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\hfill \mbox{\textit{Edexcel F1 2018 Q8 [11]}}