| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove recurrence relation formula |
| Difficulty | Challenging +1.2 This is a standard Further Maths proof by induction question with two parts: (i) proving a closed form for a recurrence relation and (ii) proving divisibility. Both are textbook applications of induction requiring multiple algebraic steps but no novel insight. Part (i) involves substituting into the recurrence relation and factoring, while part (ii) requires modular arithmetic manipulation. The question is harder than average A-level due to being Further Maths content with moderately involved algebra, but it follows completely standard templates for this topic. |
| Spec | 4.01a Mathematical induction: construct proofs |
9. (i) A sequence of numbers is defined by
$$\begin{gathered}
u _ { 1 } = 0 \quad u _ { 2 } = - 6 \\
u _ { n + 2 } = 5 u _ { n + 1 } - 6 u _ { n } \quad n \geqslant 1
\end{gathered}$$
Prove by induction that, for $n \in \mathbb { Z } ^ { + }$
$$u _ { n } = 3 \times 2 ^ { n } - 2 \times 3 ^ { n }$$
(ii) Prove by induction that, for all positive integers $n$,
$$f ( n ) = 3 ^ { 3 n - 2 } + 2 ^ { 4 n - 1 }$$
is divisible by 11\\
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\hfill \mbox{\textit{Edexcel F1 2021 Q9 [10]}}