| Exam Board | SPS |
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| Module | SPS FM (SPS FM) |
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| Year | 2025 |
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| Session | October |
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| Topic | Proof by induction |
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12. Prove by induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
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