Linear iterative formula u(n+1) = pu(n) + q - Sequences and series, recurrence and convergence

AQA C2 2006 January Q5

9 marks Moderate -0.3

5 The $n$th term of a sequence is $u _ { n }$.
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where $p$ and $q$ are constants.
The first three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$

Show that $p = 0.6$ and find the value of $q$.
Find the value of $u _ { 4 }$.
The limit of $u _ { n }$ as $n$ tends to infinity is $L$. Write down an equation for $L$ and hence find the value of $L$.

AQA C2 2008 June Q6

9 marks Moderate -0.3

6 The $n$th term of a sequence is $u _ { n }$.
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where $p$ and $q$ are constants.
The first three terms of the sequence are given by $$u _ { 1 } = - 8 \quad u _ { 2 } = 8 \quad u _ { 3 } = 4$$

Show that $q = 6$ and find the value of $p$.
Find the value of $u _ { 4 }$.
The limit of $u _ { n }$ as $n$ tends to infinity is $L$.
1. Write down an equation for $L$.
2. Hence find the value of $L$.

AQA C2 2010 June Q2

5 marks Moderate -0.8

2 The $n$th term of a sequence is $u _ { n }$.
The sequence is defined by $$u _ { n + 1 } = 6 + \frac { 2 } { 5 } u _ { n }$$ The first term of the sequence is given by $u _ { 1 } = 2$.

Find the value of $u _ { 2 }$ and the value of $u _ { 3 }$.
The limit of $u _ { n }$ as $n$ tends to infinity is $L$. Write down an equation for $L$ and hence find the value of $L$.

AQA C2 2011 June Q7

6 marks Standard +0.3

7 The $n$th term of a sequence is $u _ { n }$. The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where $p$ and $q$ are constants.
The first two terms of the sequence are given by $u _ { 1 } = 60$ and $u _ { 2 } = 48$.
The limit of $u _ { n }$ as $n$ tends to infinity is 12 .

Show that $p = \frac { 3 } { 4 }$ and find the value of $q$.
Find the value of $u _ { 3 }$.

AQA C2 2015 June Q5

6 marks Standard +0.3

5 The $n$th term of a sequence is $u _ { n }$.
The sequence is defined by $u _ { n + 1 } = p u _ { n } + q$, where $p$ and $q$ are constants.
The second term of the sequence is 160 . The third term of the sequence is 132 .
The limit of $u _ { n }$ as $n$ tends to infinity is 20 .

Find the value of $p$ and the value of $q$.
Hence find the value of the first term of the sequence.

AQA Paper 1 2023 June Q11

9 marks Standard +0.3

11 The $n$th term of a sequence is $u _ { n }$ The sequence is defined by $$u _ { n + 1 } = p u _ { n } + 70$$ where $u _ { 1 } = 400$ and $p$ is a constant.
11

Find an expression, in terms of $p$, for $u _ { 2 }$ 11
It is given that $u _ { 3 } = 382$ 11 (b) (i) Show that $p$ satisfies the equation $$200 p ^ { 2 } + 35 p - 156 = 0$$ 11 (b) (ii) It is given that the sequence is a decreasing sequence. Find the value of $u _ { 4 }$ and the value of $u _ { 5 }$ 11
The limit of $u _ { n }$ as $n$ tends to infinity is $L$ 11 (c) (i) Write down an equation for $L$ 11 (c) (ii) Find the value of $L$

AQA C2 2009 June Q3

7 marks Moderate -0.3

The $n$th term of a sequence is $u_n$. The sequence is defined by $$u_{n+1} = ku_n + 12$$ where $k$ is a constant. The first two terms of the sequence are given by $$u_1 = 16 \quad u_2 = 24$$

Show that $k = 0.75$. [2]
Find the value of $u_3$ and the value of $u_4$. [2]
The limit of $u_n$ as $n$ tends to infinity is $L$.
1. Write down an equation for $L$. [1]
2. Hence find the value of $L$. [2]

SPS SPS SM Pure 2022 June Q11

7 marks Standard +0.3

A sequence is defined by $$u_1 = 600$$ $$u_{n+1} = pu_n + q$$ where $p$ and $q$ are constants. It is given that $u_2 = 500$ and $u_4 = 356$

Find the two possible values of $u_3$ [5 marks]
When $u_n$ is a decreasing sequence, the limit of $u_n$ as $n$ tends to infinity is $L$. Write down an equation for $L$ and hence find the value of $L$. [2 marks]