# Sequences

#### Casio

##### Member
I have a linear recurrence sequence and am having a problem understanding what to do when the ratio does not seem to be the same between each of the terms, so

Terms;

4, 1.4, 2.44, 2.024.... (n = 1,2,3...)

How do I find a the ratio of these terms, and if there is none, please advise how I continue?

Kind regards

Casio #### CaptainBlack

##### Well-known member
I have a linear recurrence sequence and am having a problem understanding what to do when the ratio does not seem to be the same between each of the terms, so

Terms;

4, 1.4, 2.44, 2.024.... (n = 1,2,3...)

How do I find a the ratio of these terms, and if there is none, please advise how I continue?

Kind regards

Casio Could you tell us what the recurrence relation is?

CB

#### Casio

##### Member
Could you tell us what the recurrence relation is?

CB
U1 = 2, Un+1 =-0.3Un + 3 (n = 1,2,3...)

U2= U2.4+1 = -0.3(3.4) + 3 = 1.98

U3 = U1.98+1 = -0.3(2.98)+3 = 2.11

U4 = U2.11+1 = -0.3(3.11)+3 = 2.07

Four terms are;

2, 1.98, 2.11, 2.07,....

Not sure whether the first method I used to work out the terms was correct, or whether the method I used here is correct, the course book does not give any examples to show how they are done, only examples to solve?

Thanks

Casio

#### CaptainBlack

##### Well-known member
U1 = 2, Un+1 =-0.3Un + 3 (n = 1,2,3...)

U2= U2.4+1 = -0.3(3.4) + 3 = 1.98

U3 = U1.98+1 = -0.3(2.98)+3 = 2.11

U4 = U2.11+1 = -0.3(3.11)+3 = 2.07

Four terms are;

2, 1.98, 2.11, 2.07,....

Not sure whether the first method I used to work out the terms was correct, or whether the method I used here is correct, the course book does not give any examples to show how they are done, only examples to solve?

Thanks

Casio
What are you being asked to do with this sequence?

It obviously has an attractor at u=30/13, increasing towards it if it starts at less than 30/13 and decreasing towards it if it starts above.

CB

#### soroban

##### Well-known member
Hello, Casio!

You are misunderstanding the recurrence relation.

$$\text{Given: }\:U_{n+1} \:=\:-0.3U_n + 3,\;\;U_1 = 2$$

We have:

. . . $$\begin{array}{cccccc}U_1 &=& 2 \\ U_2 &=& -0.3(2) + 3 &=& 2.4 \\ U_3 &=& -0.3(2.4) + 3 &=& 2.28 \\ U_4 &=& -0.3(2.28) + 3 &=& 2.316 \\ \vdots && \vdots && \vdots \end{array}$$

#### Casio

##### Member
Hello, Casio!

You are misunderstanding the recurrence relation.

$$\text{Given: }\:U_{n+1} \:=\:-0.3U_n + 3,\;\;U_1 = 2$$

We have:

. . . $$\begin{array}{cccccc}U_1 &=& 2 \\ U_2 &=& -0.3(2) + 3 &=& 2.4 \\ U_3 &=& -0.3(2.4) + 3 &=& 2.28 \\ U_4 &=& -0.3(2.28) + 3 &=& 2.316 \\ \vdots && \vdots && \vdots \end{array}$$
Thanks for setting me on the right line of thought there.

OK let me take this one step at a time so I get the proper understanding of what is actually going on with these sequences.

First, please explain what this part refers to;

$$\text{Given:}\:U_{n+1}$$

Casio