Sequences and first order recurrence equations

Question 1

What is a sequence of numbers?

Answer 1

Question 2

What are the 2 ways we will start with recognising sequences?

Answer 2

sequences as explicit functions and as recurrence equations, then we will look at simple kinds of sequences such as geometric and arthemtic sequences.

Question 3

What us a sequence as an explicit function?

Answer 3

It is like a rule for the tth term of a sequence, expressed in yt.

Question 4

When ever we start looking to find terms of a sequences, we also start with what?

Answer 4

y0, which is the first term of the sequence.

Question 5

And what is the Tth term of the sequence?

Answer 5

Question 6

Answer 6

Question 7

Answer 7

y0 = 2(0) -1 = -1

y1= 2(1)-1 = 1

y2 = 2(2)-1 = 3

y3=2(3)-1 = 5

y4( =2(4)-1 = 7

yt-1 = 2(t-1)-1 = 2t-3

Question 8

What is a simple recurrence equation?

Answer 8

is an equation that defines a sequence based on a rule that gives the next term as a function of the previous term(s).

Question 9

Find the first 5 terms of this simple recurrrence equation, what is it the same as?

Answer 9

It is the same as a sequence of an explicit function yt = t+2

Question 10

Answer 10

t =2 so 2y(2-2) + y(2-1)

=2yo +y(t1) + 3

also check for the value of t

then you keep subbing in the values for t

Question 11

What is a geometric sequence, show this algerbrically and what is it as an explicit function of t?

Answer 11

A geometric sequence is found by mutliplying the previous term by a common ratio r

Question 12

Work out the first 4 terms of the geometric sequence which has its first term 2 and common ratio 3, what is the tth term of the sequence? with this geometric sequence

Answer 12

Question 13

lets say someone invests an amount P in a bank that pays compound interest at a rate 100r% per annum at the end of the year, what is the amount in the bank at the end of the first, second and 3rd year, what what is the Tth term of the sequence?

Answer 13

yt = (1+r)yt-1

this is the term term of the sequence.

Question 14

Why are geometric sequences useful in economic terms?

Answer 14

Allows us to keep track of how much investment is worth over a period of time.

Question 15

What is an arthemtic sequence and how does it look?

Answer 15

Question 16

What is the tth term of this sequence?

Answer 16

Question 17

Answer 17

Question 18

Why are arthemtic sequences useful in economic terms?

Answer 18

it allows us to see how much money we have if save a fixed amount over a period of time.

Question 19

Lets say someone places s in their piggy bank at the end of each week, what is the arthimetic sequence and what is the tth term?

Answer 19

yt=yt-1 + S for t>1 ( greater or equal)

Question 20

We have more complicated investment schemes, than arthemtic and geometric sequences, lets say for example,

write the sequence? for the end of the second month?

Answer 20

Question 21

So whats the tth term for this more complicated sequence?

Answer 21

Question 22

What do we use to solve more complicated investment schemes?

Answer 22

First order recurrrence equations.

Question 23

What is the equation for the first order linear recurrence equation?

Answer 23

Question 24

What happens to this first order recurrence equation, when b = 0 and a =1

Answer 24

Question 25

How do we solve first order recurrence equation?

Answer 25

So first of all we compare the equation we are given to the general form and make sure a doesnt equal 1 and b doesnt equal 0 2) we have to always find y\* which is our time independent solution ( it wont change) so then you have to find this by doing y\*=ay\*+b or use formula, then find general solution.

Question 26

Answer 26

Question 27

Answer 27

Question 28

Answer 28

Question 29

Returning to this complicated investment scheme solve this first order linear recurrence equation?

Answer 29

Question 30

First of all what do we notice and what can we do?

Answer 30

This is the exact same first recurrence equation as flashcard 29, so we can use it and replace, the numbers, and whenever it speaks about time, we know we are going to have to use natural logs and logarthim rules.

Question 31

Now solve this and verify your answer by subbing it back in?

Answer 31

Question 32

Answer 32

Question 33

Answer 33

Question 34

Question 3

Answer 34

Use the formula to solve, and the answers will be

Question 35

Answer 35

First of all write this out in a sequences, then find the tth term, compare it, the first order recurrence and find the expression

Question 36

q4 tbh

Question 37

q5 tbh