Why are my equations giving different curves?

[LucasGB]

What am I doing wrong?

When you have an equation such as the one displayed in Equation 1 (see attached figure) and want to compute it by hand, you may rewrite it as Equation 2.

Setting X and Y to the following values:

X = Exp[-(n-1)/beta]
Y = Exp[-T/alpha]

I got Equation 3. Thereafter, I tried to transform it in the same way, and then I got Equation 4. However, Equation 3 and 4 are not the same. For alpha=360, beta=5, A1=10 and T=1, these two equations give very different curves, as you can see at the bottom of the image (Equation 3 represented as the purple curve, and Equation 4 represented as the blue line). What am I doing wrong?

Thanks in advance,

Estevão

P.S.: Please, I really need help in this issue.

 

[uart]

Here's a hint Lucas. Go back and have a look at how you derived equation 2 from equation 1. Notice that to get it in the form of a geometric series you had to factorize out the X's and Y's across the sum of multiple terms. Think about what that means about X and Y, they must be constants, neither can depend on "n".

 

[LucasGB]

Here's a hint Lucas. Go back and have a look at how you derived equation 2 from equation 1. Notice that to get it in the form of a geometric series you had to factorize out the X's and Y's across the sum of multiple terms. Think about what that means about X and Y, they must be constants, neither can depend on "n".

You are right! Thank you very much for the answer. However, now I have a new problem: how to simplify Equation 3 so I can compute it by hand?

 

[uart]

Have you ever used Z-Transforms LucasGB? There are many ways to solve difference equations but z-transforms are one of the easiest.

See : http://en.wikipedia.org/wiki/Z-transform

I'll leave some of the work to you, but I get an answer that exactly matches your "red curve" and it's in the form of :

$$A_n = \frac{A_1}{b-a} ( b^n - a^n )$$

where "a" and "b" are constants in terms of your alpha, beta, T etc.

 

[LucasGB]

Have you ever used Z-Transforms LucasGB? There are many ways to solve difference equations but z-transforms are one of the easiest.

See : http://en.wikipedia.org/wiki/Z-transform

I'll leave some of the work to you, but I get an answer that exactly matches your "red curve" and it's in the form of :

$$A_n = \frac{A_1}{b-a} ( b^n - a^n )$$

where "a" and "b" are constants in terms of your alpha, beta, T etc.

It seems great, but I'm having a hard time searching for introdutory texts to Z-Transforms. Could you suggest one to me?

 

[Karlx]

LucasGB, I think there is no need of Z-transform.
Look at the attached picture.
I hope it will help you.
(Sorry, in the picture, when it says "knowing that X(1)=0", it should say "knowing that X(1)=1"

 

[LucasGB]

LucasGB, I think there is no need of Z-transform.
Look at the attached picture.
I hope it will help you.
(Sorry, in the picture, when it says "knowing that X(1)=0", it should say "knowing that X(1)=1"

Thank you very much, Karlx. This is a very creative way to solve the equation. You and uart helped me a lot!