Sequences and first order recurrence equations Flashcards
What is a sequence of numbers?
What are the 2 ways we will start with recognising sequences?
sequences as explicit functions and as recurrence equations, then we will look at simple kinds of sequences such as geometric and arthemtic sequences.
What us a sequence as an explicit function?
It is like a rule for the tth term of a sequence, expressed in yt.
When ever we start looking to find terms of a sequences, we also start with what?
y0, which is the first term of the sequence.
And what is the Tth term of the sequence?
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
What is a simple recurrence equation?
is an equation that defines a sequence based on a rule that gives the next term as a function of the previous term(s).
Find the first 5 terms of this simple recurrrence equation, what is it the same as?
It is the same as a sequence of an explicit function yt = t+2
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
What is a geometric sequence, show this algerbrically and what is it as an explicit function of t?
A geometric sequence is found by mutliplying the previous term by a common ratio r
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
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?
yt = (1+r)yt-1
this is the term term of the sequence.
Why are geometric sequences useful in economic terms?
Allows us to keep track of how much investment is worth over a period of time.
What is an arthemtic sequence and how does it look?
What is the tth term of this sequence?
Why are arthemtic sequences useful in economic terms?
it allows us to see how much money we have if save a fixed amount over a period of time.
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?
yt=yt-1 + S for t>1 ( greater or equal)
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?
So whats the tth term for this more complicated sequence?
What do we use to solve more complicated investment schemes?
First order recurrrence equations.
What is the equation for the first order linear recurrence equation?
What happens to this first order recurrence equation, when b = 0 and a =1
How do we solve first order recurrence equation?
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.
Returning to this complicated investment scheme solve this first order linear recurrence equation?
First of all what do we notice and what can we do?
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.
Now solve this and verify your answer by subbing it back in?
Question 3
Use the formula to solve, and the answers will be
First of all write this out in a sequences, then find the tth term, compare it, the first order recurrence and find the expression
q4 tbh
q5 tbh
