• This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. $x < - 0.414,\,\,x > 2.41$ A1A1 $\left( {x < 1 - \sqrt 2 ,\,\,x > 1 + \sqrt 2 } \right)$ Note: Award A1 for −0.414, 2.41 and A1 for correct inequalities. [2 marks]

check for $n= 3$, 16&gt; 9 so true when $n=3$ A1 assume true for $n= k$ {2^{k + 1}} &gt; {k^2} M1 Note: Award M0 for statements such as “let $n= k$”. Note: Subsequent marks after this M1 are independent of this mark and can be awarded. prove true for $n= k + 1$ ${2^{k + 2}} = 2 \times {2^{k + 1}}$ {2^{k + 2}} &gt; 2{k^2} M1 ${2^{k + 2}} = {k^2} + {k^2}$ (M1) {2^{k + 2}} &gt; {k^2} + 2k + 1 (from part (a)) A1 which is true for $k ≥ 3$ R1 Note: Only award the A1 or the R1 if it is clear why. Alternate methods are possible. ${(K + 1)^2} = \left( (K + 1) \right)^2$ hence if true for $n= k$ true for $n= k +1$, true for $n= 3$ so true for all $n≥ 3$ R1 Note: Only award the final R1 provided at least three of the previous marks are awarded. [7 marks]