Moderate -0.8 Part (i) requires checking only 4 cases (n=1,2,3,4) with simple arithmetic—no insight needed. Part (ii) is a standard proof by contradiction using parity arguments (odd/even), which is routine A-level proof technique. Both parts are straightforward applications of proof methods with minimal algebraic manipulation.
10. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\)
$$( n + 1 ) ^ { 3 } > 3 ^ { n }$$
(ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even. [0pt]
10. (i) Use proof by exhaustion to show that for $n \in \mathbb { N } , n \leqslant 4$
$$( n + 1 ) ^ { 3 } > 3 ^ { n }$$
(ii) Given that $m ^ { 3 } + 5$ is odd, use proof by contradiction to show, using algebra, that $m$ is even.\\[0pt]
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q10 [6]}}