Standard +0.8 This is a non-trivial induction proof requiring algebraic manipulation to show x_{n+1} > 2 given x_n > 2. Students must handle the rational recurrence relation carefully, requiring factorization or rearrangement of (2x_n^2 + 4x_n - 2)/(2x_n + 3) > 2, which involves clearing denominators and manipulating quadratic expressions. While the structure is standard induction, the algebraic complexity and need to work with inequalities involving fractions elevates this above routine Further Maths questions.
3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is such that \(x _ { 1 } = 3\) and
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + 4 x _ { n } - 2 } { 2 x _ { n } + 3 }$$
for \(n = 1,2,3 , \ldots\). Prove by induction that \(x _ { n } > 2\) for all \(n\).
3 The sequence $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is such that $x _ { 1 } = 3$ and
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + 4 x _ { n } - 2 } { 2 x _ { n } + 3 }$$
for $n = 1,2,3 , \ldots$. Prove by induction that $x _ { n } > 2$ for all $n$.
\hfill \mbox{\textit{CAIE FP1 2010 Q3 [6]}}