CAIE FP1 2017 Specimen — Question 4

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2017
SessionSpecimen
TopicSequences and series, recurrence and convergence

4 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that, for all positive integers \(n\), $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum \(\sum _ { n = 1 } ^ { N } a _ { n }\) is denoted by \(S _ { N }\).
  1. Find the value of \(S _ { 30 }\) correct to 3 decimal places.
  2. Find the least value of \(N\) for which \(S _ { N } > 4.9\).