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## Sunday, May 9, 2010

### Recurrence equations

Back in MST121 I learned that the solution of the following first order recurrence equation
$$a_n - 5 a_{n-1}=0$$
is
$$a_n = a_0 * 5^n.$$

I have been playing around with Mathematica and digging in Discrete Mathematics books and I am now able to solve equations of the type $a_n - c_1 \cdot a_{n-1}=f(n)$, for example if $a_0=1$ the solution of
$$a_n - 5 a_{n-1}=n^2$$
is
$$a_n = \frac{1}{32}(-8n^2-20n-15+47 \cdot 5^n).$$

Mathematica has a very nice function for solving recurrence equations ( of any order ) which is called RSolve which I only used to verify my own solution. The key to solving recurrence equations of the first order is always finding some sort of sum. Like the sum of the first $n$ integers, which is $\frac{1}{2} n(n+1)$, is in fact the solution of $a_n - a_{n-1} = n$, with $a_0=0$.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)