Prove inequality: factorial/exponential

Prove by induction that an inequality holds involving factorials or pure exponential expressions (e.g., n! > 2ⁿ, 4ⁿ > 2ⁿ + 3ⁿ, 3ⁿ > 10n) for all integers n ≥ some value.

7 questions · Standard +0.5

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CAIE FP1 2012 June Q2

5 marks Standard +0.3

2 Prove, by mathematical induction, that, for integers \(n \geqslant 2\), $$4 ^ { n } > 2 ^ { n } + 3 ^ { n }$$

OCR Further Pure Core AS 2022 June Q4

5 marks Standard +0.3

4 Prove that \(3 ^ { n } > 10 n\) for all integers \(n \geqslant 4\).

OCR Further Pure Core AS Specimen Q8

5 marks Standard +0.3

8 Prove that \(n ! > 2 ^ { n }\) for \(n \geq 4\).

OCR FP1 AS 2021 June Q4

5 marks Standard +0.3

4 Prove that \(n ! > 2 ^ { 2 n }\) for all integers \(n \geqslant 9\).

OCR Further Pure Core AS 2020 November Q6

5 marks Challenging +1.2

Prove that \(n! > 2^{2n}\) for all integers \(n \geq 9\). [5]

OCR FP1 AS 2017 December Q6

5 marks Standard +0.3

Prove by induction that \(n! \geq 6n\) for \(n \geq 4\). [5]

OCR FP1 AS 2017 Specimen Q8

5 marks Standard +0.8

Prove that \(n! > 2^n\) for \(n \geq 4\). [5]