Question 4 - A-Level Maths

AQA FP2 2012 January — Question 4 6 marks

Exam Board	AQA
Module	FP2 (Further Pure Mathematics 2)
Year	2012
Session	January
Marks	6
Paper	Download PDF ↗
Topic	Proof by induction
Type	Prove recurrence relation formula
Difficulty	Standard +0.8 This is a standard Further Maths induction proof on recurrence relations requiring algebraic manipulation to show P(n)⇒P(n+1). The substitution and simplification of the fraction (3^(n+2)-3)/(3^(n+2)-1) from the recurrence relation involves careful algebraic work with powers of 3, which is moderately challenging but follows a well-established template for FP2 students.
Spec	4.01a Mathematical induction: construct proofs

4 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { 1 } = \frac { 3 } { 4 } \quad u _ { n + 1 } = \frac { 3 } { 4 - u _ { n } }$$ Prove by induction that, for all $n \geqslant 1$, $$u _ { n } = \frac { 3 ^ { n + 1 } - 3 } { 3 ^ { n + 1 } - 1 }$$

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4 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by

$$u _ { 1 } = \frac { 3 } { 4 } \quad u _ { n + 1 } = \frac { 3 } { 4 - u _ { n } }$$

Prove by induction that, for all $n \geqslant 1$,

$$u _ { n } = \frac { 3 ^ { n + 1 } - 3 } { 3 ^ { n + 1 } - 1 }$$

\hfill \mbox{\textit{AQA FP2 2012 Q4 [6]}}

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