Recurrence relation solving for closed form

A question is this type if and only if it asks to determine a closed form (explicit formula) for a recurrence relation, typically second-order linear recurrences of the form u(n+2) = au(n+1) + bu(n) + g(n).

11 questions · Challenging +1.1

Sort by: Default | Easiest first | Hardest first
OCR MEI Further Extra Pure Specimen Q2
4 marks Challenging +1.2
2 A binary operation * is defined on the set \(S = \{ p , q , r , s , t \}\) by the following composition table.
\(*\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(q\)\(q\)\(p\)\(s\)\(t\)\(r\)
\(r\)\(r\)\(t\)\(p\)\(q\)\(s\)
\(s\)\(s\)\(r\)\(t\)\(p\)\(q\)
\(t\)\(t\)\(s\)\(q\)\(r\)\(p\)
Determine whether ( \(S , *\) ) is a group.
  1. Find the general solution of $$u _ { n } = 8 u _ { n - 1 } - 16 u _ { n - 2 } , n \geq 2 .$$ A new sequence \(v _ { n }\) is defined by \(v _ { n } = \frac { u _ { n } } { u _ { n - 1 } }\) for \(n \geq 1\).
  2. (A) Use \(( * )\) to show that \(v _ { n } = 8 - \frac { 16 } { v _ { n - 1 } }\). for \(n \geq 2\).
    (B) Deduce that if \(v _ { n }\) tends to a limit then it must be 4 .
  3. Use your general solution in part (i) to show that \(\lim _ { n \rightarrow \infty } v _ { n } = 4\).
  4. Deduce the value of \(\lim _ { n \rightarrow \infty } \left( \frac { u _ { n } } { u _ { n - 2 } } \right)\).
Edexcel FP2 2020 June Q2
9 marks Challenging +1.2
  1. Solve the recurrence system
$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4 \\ 9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$
Edexcel FP2 2021 June Q6
6 marks Challenging +1.2
  1. A recurrence system is defined by
$$\begin{aligned} u _ { n + 2 } & = 9 ( n + 1 ) ^ { 2 } u _ { n } - 3 u _ { n + 1 } \quad n \geqslant 1 \\ u _ { 1 } & = - 3 , u _ { 2 } = 18 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { N }\), $$u _ { n } = ( - 3 ) ^ { n } n !$$
Edexcel FP2 2022 June Q6
6 marks Challenging +1.2
6. (a) Determine the general solution of the recurrence relation $$u _ { n } = 2 u _ { n - 1 } - u _ { n - 2 } + 2 ^ { n } \quad n \geqslant 2$$ (b) Hence solve this recurrence relation given that \(u _ { 0 } = 2 u _ { 1 }\) and \(u _ { 4 } = 3 u _ { 2 }\)
Edexcel FP2 2023 June Q6
7 marks Challenging +1.2
  1. Determine a closed form for the recurrence relation
$$\begin{aligned} & u _ { 0 } = 1 \quad u _ { 1 } = 4 \\ & u _ { n + 2 } = 2 u _ { n + 1 } - \frac { 4 } { 3 } u _ { n } + n \quad n \geqslant 0 \end{aligned}$$
Edexcel FP2 2024 June Q2
5 marks Standard +0.3
  1. Determine a closed form for the recurrence system
$$\begin{gathered} u _ { 1 } = 4 \quad u _ { 2 } = 6 \\ u _ { n + 2 } = 6 u _ { n + 1 } - 9 u _ { n } \quad ( n = 1,2,3 , \ldots ) \end{gathered}$$
Edexcel FD2 2021 June Q4
11 marks Challenging +1.2
  1. Sequences \(\left\{ x _ { n } \right\}\) and \(\left\{ y _ { n } \right\}\) for \(n \in \mathbb { N }\), are defined by
$$\begin{gathered} x _ { n + 1 } = 2 y _ { n } + 3 \quad \text { and } \quad y _ { n + 1 } = 3 x _ { n + 1 } - 4 x _ { n } \\ x _ { 1 } = 1 \quad \text { and } \quad y _ { 1 } = a \end{gathered}$$ where \(a\) is a constant.
  1. Show that \(x _ { n + 2 } - 6 x _ { n + 1 } + 8 x _ { n } = 3\)
  2. Solve the second-order recurrence relation given in (a) to obtain an expression for \(x _ { n }\) in terms of \(a\) and \(n\). Given that \(x _ { 7 } = 28225\)
  3. find the value of \(a\).
Edexcel FD2 2023 June Q5
8 marks Challenging +1.2
5. A sequence \(\left\{ u _ { n } \right\}\), where \(\mathrm { n } \geqslant 0\), satisfies the second order recurrence relation $$u _ { n + 2 } = \frac { 1 } { 2 } \left( u _ { n + 1 } + u _ { n } \right) + 3 \text { where } u _ { 0 } = 15 \quad u _ { 1 } = 20$$
  1. By considering the sequence \(\left\{ v _ { n } \right\}\), where \(u _ { n } = v _ { n } + 2 n\) for \(\mathrm { n } \geqslant 0\), determine an expression for \(u _ { n }\) as a function of n .
  2. Describe the long-term behaviour of \(u _ { n }\)
Edexcel FD2 2024 June Q8
8 marks Challenging +1.2
8. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation $$2 u _ { n + 2 } + 5 u _ { n + 1 } = 3 u _ { n } + 8 n + 2$$
  1. Find the general solution of this recurrence relation.
    (5) A particular solution of this recurrence relation has \(u _ { 0 } = 1\) and \(u _ { 1 } = k\), where \(k\) is a positive constant. All terms of the sequence are positive.
  2. Determine the value of \(k\).
    (3)
Edexcel FD2 Specimen Q1
6 marks Standard +0.8
  1. (a) Find the general solution of the recurrence relation
$$u _ { n + 2 } = u _ { n + 1 } + u _ { n } , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 1\) and \(u _ { 2 } = 1\) (b) find the particular solution of the recurrence relation.
OCR Further Additional Pure 2018 September Q7
14 marks Challenging +1.8
7 The members of the family of the sequences \(\left\{ u _ { n } \right\}\) satisfy the recurrence relation $$u _ { n + 1 } = 10 u _ { n } - u _ { n - 1 } \text { for } n \geqslant 1$$
  1. Determine the general solution of (*).
  2. The sequences \(\left\{ a _ { n } \right\}\) and \(\left\{ b _ { n } \right\}\) are members of this family of sequences, corresponding to the initial terms \(a _ { 0 } = 1 , a _ { 1 } = 5\) and \(b _ { 0 } = 0 , b _ { 1 } = 2\) respectively.
    (a) Find the next two terms of each sequence.
    (b) Prove that, for all non-negative integers \(n , \left( a _ { n } \right) ^ { 2 } - 6 \left( b _ { n } \right) ^ { 2 } = 1\).
    (c) Determine \(\lim _ { n \rightarrow \infty } \left( \frac { a _ { n } } { b _ { n } } \right)\). \section*{OCR} Oxford Cambridge and RSA