Recurrence relation evaluation

Given a recurrence relation u_(n+1) = f(u_n), find specific terms or sums by iterative calculation.

63 questions · Moderate -0.6

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Edexcel C12 2015 January Q8
9 marks Moderate -0.8
  1. A sequence is defined by
$$\begin{aligned} u _ { 1 } & = k \\ u _ { n + 1 } & = 3 u _ { n } - 12 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down fully simplified expressions for \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\) in terms of \(k\). Given that \(u _ { 4 } = 15\)
  2. find the value of \(k\),
  3. find \(\sum _ { i = 1 } ^ { 4 } u _ { i }\), giving an exact numerical answer.
Edexcel C12 2016 January Q1
5 marks Easy -1.2
  1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies
$$u _ { n + 1 } = 2 u _ { n } - 6 , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 2\)
  1. find the value of \(u _ { 3 }\)
  2. evaluate \(\sum _ { i = 1 } ^ { 4 } u _ { i }\)
Edexcel C12 2018 January Q2
5 marks Moderate -0.8
2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = 2 - 3 u _ { n } \quad n \geqslant 1 \end{aligned}$$
  1. Find the value of \(u _ { 2 }\) and the value of \(u _ { 3 }\)
  2. Calculate the value of \(\sum _ { r = 1 } ^ { 4 } \left( r - u _ { r } \right)\) □
Edexcel C12 2019 January Q4
6 marks Moderate -0.8
4. A sequence is defined by $$\begin{aligned} u _ { 1 } & = k , \text { where } k \text { is a constant } \\ u _ { n + 1 } & = 4 u _ { n } - 3 , n \geqslant 1 \end{aligned}$$
  1. Find \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\), simplifying your answers as appropriate. Given \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 18\)
  2. find \(k\).
Edexcel C12 2016 June Q5
6 marks Standard +0.3
5. (i) $$U _ { n + 1 } = \frac { U _ { n } } { U _ { n } - 3 } , \quad n \geqslant 1$$ Given \(U _ { 1 } = 4\), find
  1. \(U _ { 2 }\)
  2. \(\sum _ { n = 1 } ^ { 100 } U _ { n }\) (ii) Given $$\sum _ { r = 1 } ^ { n } ( 100 - 3 r ) < 0$$ find the least value of the positive integer \(n\).
Edexcel C12 2018 June Q7
8 marks Easy -1.3
7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1 \end{aligned}$$ Find the values of
  1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. \(u _ { 100 }\)
  3. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)
Edexcel C12 2019 June Q5
6 marks Standard +0.3
5. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = k - \frac { 8 } { u _ { n } } \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given that \(u _ { 3 } = 6\)
  2. find the possible values of \(k\).
Edexcel C12 Specimen Q7
5 marks Moderate -0.3
7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { 1 } = 2 \\ a _ { n + 1 } = 3 a _ { n } - c \end{gathered}$$ where \(c\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
  2. find the value of \(c\).
Edexcel C1 2008 January Q7
8 marks Moderate -0.8
  1. A sequence is given by:
$$\begin{aligned} & x _ { 1 } = 1 \\ & x _ { n + 1 } = x _ { n } \left( p + x _ { n } \right) \end{aligned}$$ where \(p\) is a constant ( \(p \neq 0\) ) .
  1. Find \(x _ { 2 }\) in terms of \(p\).
  2. Show that \(x _ { 3 } = 1 + 3 p + 2 p ^ { 2 }\). Given that \(x _ { 3 } = 1\),
  3. find the value of \(p\),
  4. write down the value of \(x _ { 2008 }\).
Edexcel C1 2011 January Q4
5 marks Moderate -0.5
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 2 \\ a _ { n + 1 } & = 3 a _ { n } - c \end{aligned}$$ where \(c\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
  2. find the value of \(c\).
Edexcel C1 2012 January Q4
6 marks Moderate -0.8
4. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{aligned} x _ { 1 } & = 1 \\ x _ { n + 1 } & = a x _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(a\) is a constant.
  1. Write down an expression for \(x _ { 2 }\) in terms of \(a\).
  2. Show that \(x _ { 3 } = a ^ { 2 } + 5 a + 5\) Given that \(x _ { 3 } = 41\)
  3. find the possible values of \(a\).
Edexcel C1 2013 January Q4
5 marks Moderate -0.8
4. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n + 1 } = 2 u _ { n } - 1 , n \geqslant 1$$ Given that \(u _ { 2 } = 9\),
  1. find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\),
  2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\).
Edexcel C1 2006 June Q4
5 marks Moderate -0.8
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3 \\ & a _ { n + 1 } = 3 a _ { n } - 5 , \quad n \geqslant 1 . \end{aligned}$$
  1. Find the value of \(a _ { 2 }\) and the value of \(a _ { 3 }\).
  2. Calculate the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\).
Edexcel C1 2007 June Q8
7 marks Moderate -0.8
8. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k \\ a _ { n + 1 } & = 3 a _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a positive integer.
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  2. Show that \(a _ { 3 } = 9 k + 20\).
    1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\).
    2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 10 .
Edexcel C1 2008 June Q5
6 marks Moderate -0.8
5. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 , \\ x _ { n + 1 } = a x _ { n } - 3 , n \geqslant 1 , \end{gathered}$$ where \(a\) is a constant.
  1. Find an expression for \(x _ { 2 }\) in terms of \(a\).
  2. Show that \(x _ { 3 } = a ^ { 2 } - 3 a - 3\). Given that \(x _ { 3 } = 7\),
  3. find the possible values of \(a\).
Edexcel C1 2009 June Q7
7 marks Moderate -0.3
7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k \\ a _ { n + 1 } & = 2 a _ { n } - 7 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  2. Show that \(a _ { 3 } = 4 k - 21\). Given that \(\sum _ { r = 1 } ^ { 4 } a _ { r } = 43\),
  3. find the value of \(k\).
Edexcel C1 2010 June Q5
4 marks Moderate -0.8
  1. A sequence of positive numbers is defined by
$$\begin{aligned} a _ { n + 1 } & = \sqrt { } \left( a _ { n } ^ { 2 } + 3 \right) , \quad n \geqslant 1 , \\ a _ { 1 } & = 2 \end{aligned}$$
  1. Find \(a _ { 2 }\) and \(a _ { 3 }\), leaving your answers in surd form.
  2. Show that \(a _ { 5 } = 4\)
Edexcel C1 2011 June Q5
7 marks Moderate -0.8
5. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k \\ a _ { n + 1 } & = 5 a _ { n } + 3 , \quad n \geqslant 1 , \end{aligned}$$ where \(k\) is a positive integer.
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  2. Show that \(a _ { 3 } = 25 k + 18\).
    1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\), in its simplest form.
    2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 6 .
Edexcel C1 2012 June Q5
7 marks Moderate -0.3
5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3 \\ & a _ { n + 1 } = 2 a _ { n } - c \quad ( n \geqslant 1 ) \end{aligned}$$ where \(c\) is a constant.
  1. Write down an expression, in terms of \(c\), for \(a _ { 2 }\)
  2. Show that \(a _ { 3 } = 12 - 3 c\) Given that \(\sum _ { i = 1 } ^ { 4 } a _ { i } \geqslant 23\)
  3. find the range of values of \(c\).
Edexcel C1 2013 June Q6
9 marks Moderate -0.5
6. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 \\ x _ { n + 1 } = \left( x _ { n } \right) ^ { 2 } - k x _ { n } , \quad n \geqslant 1 \end{gathered}$$ where \(k\) is a constant, \(k \neq 0\)
  1. Find an expression for \(x _ { 2 }\) in terms of \(k\).
  2. Show that \(x _ { 3 } = 1 - 3 k + 2 k ^ { 2 }\) Given also that \(x _ { 3 } = 1\),
  3. calculate the value of \(k\).
  4. Hence find the value of \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)
Edexcel C1 2013 June Q4
7 marks Standard +0.3
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = k \left( a _ { n } + 2 \right) , \quad \text { for } n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 2\),
  2. find the two possible values of \(k\).
Edexcel C1 2016 June Q6
6 marks Moderate -0.3
6. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = 5 - k a _ { n } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\). Find
  2. \(\sum _ { r = 1 } ^ { 3 } \left( 1 + a _ { r } \right)\) in terms of \(k\), giving your answer in its simplest form,
  3. \(\sum _ { r = 1 } ^ { 100 } \left( a _ { r + 1 } + k a _ { r } \right)\)
Edexcel C1 2017 June Q3
6 marks Moderate -0.3
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned} a _ { 1 } & = 1 \\ a _ { n + 1 } & = \frac { k \left( a _ { n } + 1 \right) } { a _ { n } } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a positive constant.
  1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\), giving your answers in their simplest form. Given that \(\sum _ { r = 1 } ^ { 3 } a _ { r } = 10\)
  2. find an exact value for \(k\).
Edexcel C1 2018 June Q6
7 marks Moderate -0.8
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = \frac { a _ { n } } { a _ { n } + 1 } , \quad n \geqslant 1 , n \in \mathbb { N } \end{aligned}$$
  1. Find the values of \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\) Write your answers as simplified fractions. Given that $$a _ { n } = \frac { 4 } { p n + q } , \text { where } p \text { and } q \text { are constants }$$
  2. state the value of \(p\) and the value of \(q\).
  3. Hence calculate the value of \(N\) such that \(a _ { N } = \frac { 4 } { 321 }\)
Edexcel P2 2021 October Q2
5 marks Moderate -0.5
2. A sequence is defined by