## Sunday, 14 March 2010

### Progress: Basic Algebraic Skills V3.8

Ch.7 Equations <- nearly complete
I'm thinking whether I should also put the convergence of series into this chapter since it may contain differentiation again~
Harmonic series as a special case of zeta function will surely be an interesting topic, but it must be far too difficult... Maybe we can also talk about the big O function? lol
I'm trying to show that the characteristic equation in the third degree is somehow vaild as an extension of cubic equation, but it's also too difficult.
One thing that I have to mention is that I don't really have something called skills to share about. Considering HCM's (not acid) notes, the only thing I'll discuss is the transformation of system and technique of substitution. For example, the geometry idea of inversion can be applied into the co-goem section (but the transformation of graph is too hard Orz).

A sample of recurrence equation problem
1)It's given that a_(n+2)=a_(n+1)+ka_(n). If the sum of 10 consecutive terms is equal to 11 times the seventh term, find k.

Solution:
It results in a quartic equation, though the answer is obvious.
k^4+15k^3+35k^2+28k+9=11(3k^2+4k+1)...

2)Given a sequence a_(n+2)=a_(n+1)+2a_(n). If a_(n)=p_(n)a_(1)+q_(n)a_(2), show that |p_(n)-q_(n)|=1. Hence show that p/q = O(1).

This solution is left for you =)
Hint: we can try to express p not just only in terms of other p, but also q. If we can link them together, the problem must be solved.