Second-Order Non-Homogeneous Recurrence Relations

A question is this type if and only if it involves solving second-order recurrence relations with a non-zero right-hand side by finding complementary function and particular integral.

2 questions · Challenging +1.8

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OCR MEI Further Extra Pure 2024 June Q2
12 marks Challenging +1.8
2
  1. Determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 0\).
  2. Using your answer to part (a), determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 20 n ^ { 2 } + 60 n\). In the rest of this question the sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation in part (b). You are given that \(u _ { 0 } = - 9\) and \(u _ { 1 } = - 12\).
  3. Determine the particular solution for \(\mathrm { u } _ { \mathrm { n } }\). You are given that, as \(n\) increases, once the values of \(u _ { n }\) start to increase, then from that point onwards the sequence is an increasing sequence.
  4. Use your answer to part (c) to determine, by direct calculation, the least value taken by terms in the sequence. You should show any values that you rely on in your argument.
OCR MEI Further Extra Pure 2021 November Q4
14 marks Challenging +1.8
4 The sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation \(u _ { n + 2 } - 3 u _ { n + 1 } - 10 u _ { n } = 24 n - 10\).
  1. Determine the general solution of the recurrence relation.
  2. Hence determine the particular solution of the recurrence relation for which \(u _ { 0 } = 6\) and \(u _ { 1 } = 10\).
  3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u _ { 2 }\). The sequence \(v _ { 0 } , v _ { 1 } , v _ { 2 } , \ldots\) is defined by \(v _ { n } = \frac { u _ { n } } { p ^ { n } }\) for some constant \(p\), where \(u _ { n }\) denotes the
    particular solution found in part (b). particular solution found in part (b). You are given that \(\mathrm { v } _ { \mathrm { n } }\) converges to a finite non-zero limit, \(q\), as \(n \rightarrow \infty\).
  4. Determine \(p\) and \(q\).