| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove divisibility |
| Difficulty | Standard +0.8 This is a Further Maths proof by induction requiring students to handle two different bases (4 and 5) with different exponents, and manipulate the algebraic expression to factor out 21. While the inductive step follows standard technique, the algebraic manipulation (rewriting 4^(k+3) + 5^(2k+3) to reveal the factor of 21) requires more sophistication than typical single-base divisibility proofs, placing it moderately above average difficulty. |
| Spec | 4.01a Mathematical induction: construct proofs |
8. Prove by induction that $4 ^ { n + 2 } + 5 ^ { 2 n + 1 }$ is divisible by 21 for all positive integers $n$.\\
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\hfill \mbox{\textit{Edexcel F1 2021 Q8 [6]}}