Simplifying Equations: A Scientist's Guide

[LucasGB]

I have the following equations:

(See Figure 1)

I tried summing them up in one single general formula, and I got:

(See Figure 2)

I'd like to know two things:

1. Is my general formula correct? Please note it does not work for A1, since for my purposes, A0 does not exist.

2. If my formula is correct, how can I make it simpler, so I can computate An by hand?

Thank you very much for your help! I'm sure this is going to be simple, but my knowledge is limited.

 

[ross_tang]

1. Your equation is indeed correct. But I prefer to write it this way:
$$A_n=A_{n-1}y+A_1x$$

2. If you want to make it simple and compute it by hand, I guess you would like to solve the above difference equation. Here is the solution:

$$A_n=A_1y^{n-1}+A_1x\left(\frac{y^{n-1}-1}{y-1}\right)$$

Please refer to this http://www.voofie.com/content/77/general-term-of-recurrence-relation-and-its-solution/" , if you are interested in how to get the answer. With this equation, I think you can compute any term by hand easily.

 

[LucasGB]

Thank you very much, that was very helpful!

 

[ross_tang]

You are welcome. Is it some kind of homework problem? Or you encounter it in your self-study?

 

[LucasGB]

Just a problem that arose while studying some experiments. Not homework related. Thank you again.