How to set up a system of linear equations with loop and solve it?

There is a system of linear equations with 1000 equations and 1000 variables. The 1000 variables are: x(1),x(2),...,x(999),b. The 1000 equations are: 6*x(i)+(1000-i)*(x(i+1)-x(i))-5*x(i+500)=1 (i=0,1,...,499), and x(i)=1+b*(2/3-i*2/3000)^6 (i=500,501,...,999), where x(0)=0.8. How to program this system of linear equations and solve it?

[dpb]

Can't interpret the form, sorry...write out a few explicitly to illustrate and maybe somebody can figure out the pattern from that.

[Chong Zhang]

I revised it. Can you read it now?

[Torsten]

If you understood the solution of the two ODEs that are coupled at x=1/3, you should also come up with a solution of this linear system of equations that is coupled at i=500.

Hint:

The "real" unknown is b which can be determined by making x(500) from the first recursion equal x(500) from the second recursion.

Best wishes

Torsten.