| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | January |
| Marks | 12 |
| 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 closed form for a recurrence relation using straightforward algebraic manipulation, while part (ii) is a routine divisibility proof. Both follow textbook templates with no novel insights required, making this slightly easier than average even for Further Maths students. |
| Spec | 4.01a Mathematical induction: construct proofs |
| VIXV SIHIANI III IM IONOO | VIAV SIHI NI JYHAM ION OO | VI4V SIHI NI JLIYM ION OO |
9. (i) A sequence of numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by
$$u _ { n + 1 } = \frac { 1 } { 3 } \left( 2 u _ { n } - 1 \right) \quad u _ { 1 } = 1$$
Prove by induction that, for $n \in \mathbb { Z } ^ { + }$
$$u _ { n } = 3 \left( \frac { 2 } { 3 } \right) ^ { n } - 1$$
(ii) $\mathrm { f } ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$
Prove by induction that, for $n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )$ is a multiple of 7\\
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VIXV SIHIANI III IM IONOO & VIAV SIHI NI JYHAM ION OO & VI4V SIHI NI JLIYM ION OO \\
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\hfill \mbox{\textit{Edexcel F1 2021 Q9 [12]}}