Standard +0.3 This is a straightforward induction proof with a given formula to verify. The recurrence relation substitution is algebraic but routine, requiring only fraction manipulation and simplification. While it's a Further Maths topic, the mechanics are standard: verify base case, assume for n=k, prove for n=k+1 by substituting into the recurrence relation. No novel insight or complex algebraic manipulation required beyond typical A-level techniques.
2 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 1\) and \(u _ { r + 1 } = \frac { 3 u _ { r } - 2 } { 4 }\) for all \(r\). Prove by mathematical induction that \(u _ { n } = 4 \left( \frac { 3 } { 4 } \right) ^ { n } - 2\), for all positive integers \(n\).
2 For the sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$, it is given that $u _ { 1 } = 1$ and $u _ { r + 1 } = \frac { 3 u _ { r } - 2 } { 4 }$ for all $r$. Prove by mathematical induction that $u _ { n } = 4 \left( \frac { 3 } { 4 } \right) ^ { n } - 2$, for all positive integers $n$.
\hfill \mbox{\textit{CAIE FP1 2012 Q2 [5]}}