Pre-U Pre-U 9795/1 2016 Specimen — Question 5

Exam BoardPre-U
ModulePre-U 9795/1 (Pre-U Further Mathematics Paper 1)
Year2016
SessionSpecimen
TopicSequences and series, recurrence and convergence
TypeMethod of differences with given identity
DifficultyStandard +0.3 This is a straightforward proof by induction with a given result to prove. The base case is trivial arithmetic, and the inductive step requires standard algebraic manipulation of fractions with a clear path forward. While it involves some careful algebra, it's a routine application of the induction technique with no conceptual obstacles or novel insights required—slightly easier than average for A-level.
Spec4.01a Mathematical induction: construct proofs
5 Use induction to prove that \(\sum _ { r = 1 } ^ { n } \left( \frac { 2 } { 4 r - 1 } \right) \left( \frac { 2 } { 4 r + 3 } \right) = \frac { 1 } { 3 } - \frac { 1 } { 4 n + 3 }\) for all positive integers \(n\). 
5 Use induction to prove that $\sum _ { r = 1 } ^ { n } \left( \frac { 2 } { 4 r - 1 } \right) \left( \frac { 2 } { 4 r + 3 } \right) = \frac { 1 } { 3 } - \frac { 1 } { 4 n + 3 }$ for all positive integers $n$.

\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q5}}
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