## Question 15(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $n=1$: $2^3=8$, $3^1=3$, $(8>3)$ | M1 | 2.1 |
| $n=2$: $3^3=27$, $3^2=9$, $(27>9)$ | | |
| $n=3$: $4^3=64$, $3^3=27$, $(64>27)$ | | |
| $n=4$: $5^3=125$, $3^4=81$, $(125>81)$ | | |
| So if $n \leqslant 4, n \in \mathbb{N}$ then $(n+1)^3 > 3^n$ | A1 | 2.4 |

**Notes:**
- M1: Full and rigorous argument using all of $n=1,2,3$ and $4$. Award for attempts at both $(n+1)^3$ and $3^n$ for **ALL** values with at least 5 of 8 values correct. No requirement to compare sizes e.g. stating $27>9$. Extra values e.g. $n=0$ may be ignored.
- A1: All values for $n=1,2,3,4$ correct, all pairs compared correctly, and a minimal conclusion. Accept $\checkmark$ or "hence proven"

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## Question 15(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Begins proof by negating the statement: "Let $m$ be odd" or "Assume $m$ is not even" | M1 | 2.4 |
| Set $m=(2p\pm1)$ and attempt $m^3+5=(2p\pm1)^3+5=\ldots$ | M1 | 2.1 |
| $=8p^3+12p^2+6p+6$ AND deduces even | A1 | 2.2a |
| Completes proof with reason and conclusion: reason for $8p^3+12p^2+6p+6$ being even; acceptable statement such as "this is a contradiction so if $m^3+5$ is odd then $m$ must be even" | A1 | 2.4 |

**Notes:**
- M1(first): Cannot be scored if candidate attempts $m$ both odd and even
- M1(second): Award for a 4-term cubic expression
- A1(first): Correctly reaches $(2p+1)^3+5=8p^3+12p^2+6p+6$ and **states** even. Alternatively $(2p-1)^3+5=8p^3-12p^2+6p+4$ and **states** even
- A1(second): Requires **(1) a reason** why $8p^3+12p^2+6p+6$ or $8p^3-12p^2+6p+4$ is even (e.g. all terms are even, or factorised form e.g. $8p^3-12p^2+6p+4=2(4p^3-6p^2+3p+2)$) AND **(2)** an acceptable concluding statement such as "this is a contradiction, so if $m^3+5$ is odd then $m$ is even", or "this is contradiction, so proven", or "So if $m^3+5$ is odd then $m$ is even"
- S.C: If candidate misinterprets and uses a counterexample to "if $m^3+5$ is odd then $m$ must be even" such as $m=\sqrt[3]{2}$, they can score special case mark **B1**