| Exam Board | SPS |
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| Module | SPS FM Pure (SPS FM Pure) |
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| Year | 2024 |
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| Session | February |
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| Marks | 6 |
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| Topic | Proof by induction |
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5. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$u _ { 1 } = 0 \quad u _ { n + 1 } = \frac { 5 } { 6 - u _ { n } }$$
Prove by induction that, for all integers \(n \geq 1\),
$$u _ { n } = \frac { 5 ^ { n } - 5 } { 5 ^ { n } - 1 }$$
[6 marks]
[0pt]
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