Simple recurrence evaluation

A question is this type if and only if it defines a simple first-order recurrence relation u(n+1) = f(u(n)) with explicit formula and asks to find specific terms or sum the sequence, without requiring solving for closed form or analyzing convergence.

14 questions · Moderate -0.1

1.04e Sequences: nth term and recurrence relations

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Edexcel P2 2019 June Q1

4 marks Easy -1.3

A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by

$$\begin{aligned} a _ { n + 1 } & = 4 - a _ { n } \\ a _ { 1 } & = 3 \end{aligned}$$ Find the value of

1. $a _ { 2 }$
2. $a _ { 107 }$
$\sum _ { n = 1 } ^ { 200 } \left( 2 a _ { n } - 1 \right)$

Edexcel P2 2023 October Q2

5 marks Moderate -0.3

A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by

$$\begin{gathered} u _ { 1 } = 3 \\ u _ { n + 1 } = 2 - \frac { 4 } { u _ { n } } \end{gathered}$$

Find the value of $u _ { 2 }$, the value of $u _ { 3 }$ and the value of $u _ { 4 }$
Find the value of $$\sum _ { r = 1 } ^ { 100 } u _ { r }$$

Edexcel Paper 2 2018 June Q4

7 marks Moderate -0.3

(i) Show that $\sum _ { r = 1 } ^ { 16 } \left( 3 + 5 r + 2 ^ { r } \right) = 131798$ (ii) A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by

$$u _ { n + 1 } = \frac { 1 } { u _ { n } } , \quad u _ { 1 } = \frac { 2 } { 3 }$$ Find the exact value of $\sum _ { r = 1 } ^ { 100 } u _ { r }$

OCR MEI Paper 3 2022 June Q3

4 marks Moderate -0.3

3 An infinite sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by $a _ { \mathrm { n } } = \frac { \mathrm { n } } { \mathrm { n } + 1 }$, for all positive integers $n$.

Find the limit of the sequence.
Prove that this is an increasing sequence.

OCR Further Additional Pure AS 2018 March Q3

5 marks Standard +0.3

3 In this question you must show detailed reasoning.

The sequence $\left\{ A _ { n } \right\}$ is given by $A _ { 1 } = \sqrt { 3 }$ and $A _ { n + 1 } = ( \sqrt { 3 } + 1 ) A _ { n }$ for $n \geqslant 1$. Find an expression for $A _ { n }$ in terms of $n$.
The sequence $\left\{ B _ { n } \right\}$ is given by the formula $$B _ { n } = \frac { 1 } { \sqrt { 3 } } \left( ( \sqrt { 3 } + 1 ) ^ { n } - ( \sqrt { 3 } - 1 ) ^ { n } \right) \text { for } n \geqslant 1 .$$ Explain why $B _ { n } \rightarrow \frac { 1 } { \sqrt { 3 } } ( \sqrt { 3 } + 1 ) ^ { n }$ as $n \rightarrow \infty$.
The sequence $\left\{ C _ { n } \right\}$ converges and is defined by $C _ { n } = \frac { A _ { n } } { B _ { n } }$ for $n \geqslant 1$. Identify the limit of $C n$ as $n \rightarrow \infty$.

AQA Paper 1 2020 June Q7

4 marks Moderate -0.8

7 Consecutive terms of a sequence are related by $$u _ { n + 1 } = 3 - \left( u _ { n } \right) ^ { 2 }$$ 7

In the case that $u _ { 1 } = 2$ 7
1. (i) Find $u _ { 3 }$ 7
2. (ii) Find $u _ { 50 }$ 7
3. State a different value for $u _ { 1 }$ which gives the same value for $u _ { 50 }$ as found in part (a)(ii).

Edexcel PURE 2024 October Q2

Standard +0.8

A sequence of numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by

$$\begin{gathered} u _ { 1 } = 7 \\ u _ { n + 1 } = ( - 1 ) ^ { n } u _ { n } + k \end{gathered}$$ where $k$ is a constant.

Show that $u _ { 5 } = 7$ Given that $\sum _ { r = 1 } ^ { 4 } u _ { r } = 30$
find the value of $k$.
Hence find the value of $\sum _ { r = 1 } ^ { 150 } u _ { r }$

Pre-U Pre-U 9794/2 2013 June Q2

7 marks Easy -1.3

An arithmetic sequence has first term 3 and common difference 2. Find the twenty-first term of this sequence.
Find the sum to infinity of a geometric progression with first term 162 and second term 54.
A sequence is given by the recurrence relation $u _ { 1 } = 3 , u _ { n + 1 } = 2 - u _ { n } , n = 1,2,3 , \ldots$. Find $u _ { 2 } , u _ { 3 }$, $u _ { 4 } , u _ { 5 }$ and describe the behaviour of this sequence.