Recursive sequence definition

A question is this type if and only if a geometric sequence is defined recursively (e.g., u_1 = a, u_{n+1} = r·u_n) and you must find specific terms, sums, or other properties.

7 questions · Moderate -0.8

1.04i Geometric sequences: nth term and finite series sum 1.04j Sum to infinity: convergent geometric series |r|<1

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Edexcel C12 2015 June Q10

8 marks Moderate -0.3

10. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 4 \\ u _ { n + 1 } & = \frac { 2 u _ { n } } { 3 } , \quad n \geqslant 1 \end{aligned}$$

Find the exact values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$
Find the value of $u _ { 20 }$, giving your answer to 3 significant figures.
Evaluate $$12 - \sum _ { i = 1 } ^ { 16 } u _ { i }$$ giving your answer to 3 significant figures.
Explain why $\sum _ { i = 1 } ^ { N } u _ { i } < 12$ for all positive integer values of $N$.

Edexcel C12 2016 October Q6

9 marks Easy -1.2

6. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 36 \\ u _ { n + 1 } & = \frac { 2 } { 3 } u _ { n } , \quad n \geqslant 1 \end{aligned}$$

Find the exact simplified values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$
Write down the common ratio of the sequence.
Find, giving your answer to 4 significant figures, the value of $u _ { 11 }$
Find the exact value of $\sum _ { i = 1 } ^ { 6 } u _ { i }$
Find the value of $\sum _ { i = 1 } ^ { \infty } u _ { i }$

OCR C2 2007 June Q1

5 marks Easy -1.2

1 A geometric progression $\mathrm { u } _ { 1 } , \mathrm { u } _ { 2 } , \mathrm { u } _ { 3 } , \ldots$ is defined by $$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 .$$

Write down the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
Find $\sum _ { n = 1 } ^ { 20 } u _ { n }$.

OCR MEI C2 Q2

5 marks Moderate -0.8

2 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 , \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } . \end{aligned}$$

Find the third term of this sequence and state what type of sequence it is.
Show that the series $u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$ converges and find its sum to infinity.

OCR MEI C2 2009 January Q8

5 marks Moderate -0.8

8 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } \end{aligned}$$

Find the third term of this sequence and state what type of sequence it is.
Show that the series $u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$ converges and find its sum to infinity.

OCR H240/01 2018 December Q6

10 marks Moderate -0.3

6 In this question you must show detailed reasoning.
A sequence $S$ has terms $u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots$ defined by $u _ { 1 } = 500$ and $u _ { n + 1 } = 0.8 u _ { n }$.

State whether $S$ is an arithmetic sequence or a geometric sequence, giving a reason for your answer.
Find $u _ { 20 }$.
Find $\sum _ { n = 1 } ^ { 20 } u _ { n }$.
Given that $\sum _ { n = k } ^ { \infty } u _ { n } = 1024$, find the value of $k$.

Pre-U Pre-U 9794/2 2018 June Q1

4 marks Easy -1.2

1 A geometric progression $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { 1 } = 32$ and $u _ { n + 1 } = 0.75 u _ { n }$ for $n \geqslant 1$.

Find $u _ { 5 }$.
Find $\sum _ { n = 1 } ^ { \infty } u _ { n }$.