| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2012 |
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| Session | June |
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| Topic | Proof by induction |
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2 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 1\) and \(u _ { r + 1 } = \frac { 3 u _ { r } - 2 } { 4 }\) for all \(r\). Prove by mathematical induction that \(u _ { n } = 4 \left( \frac { 3 } { 4 } \right) ^ { n } - 2\), for all positive integers \(n\).