The characteristic equation of is so, its solutions are . A particular solution of the complete has the form ... etc .
I think this is the right section since sequences belong to discrete math. Anyway, I have this sequence and I need to know if there is any way to remove the recursion from it:
I tried to somehow turn it into exponential function but I don't know how, then I thought about moving something to the other side but I don't think it would work either. Is it even possible to write it without recursion?
Thank you so much for your help!
However, there are still few lines that I don't quite understand and I'd appreciate a little further explanation because I'm not really this good with math...
I know that here we calculate the difference between any two neighbouring elements of the sequence...
Now me move everything to the left, create second equation with incremented variable and substract the left sides to get another equation...
Here's the first thing I don't get - why do we turn n-th element of the sequence into n-th power of X?
I understand that here we divide every element by X^(n-1) to simplify the equation as much as possible...
Now we figure out the three possible values of X in this equation...
And here's my other problem. Why do we raise the values of X to n-th power? Why is this function a sum of these powers anyway? And why is B multiplied by n, but A and C are not?
Okay, the rest is obvious - we figure out the value of A, B and C and we get the function.
Sorry for the lack of correct English terms here and there.
See this theorem about the solution of a linear homogeneous recurrence relation with constant coefficients.