| Exam Board | Edexcel |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2011 |
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| Session | January |
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| Topic | Proof by induction |
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9. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , u _ { 4 } , \ldots\) is defined by
$$u _ { n + 1 } = 4 u _ { n } + 2 , \quad u _ { 1 } = 2$$
Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$u _ { n } = \frac { 2 } { 3 } \left( 4 ^ { n } - 1 \right)$$