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## Saturday, May 1, 2010

### Exercise

I would like to propose a challenge.

What follows ...
a) 1 - 2 - 6 - 15 - 40 - 104 - 273 - ?
b) If the sequence in a) is generated by a function f: N->R then what is f[n] ? Or... what is the closed form for the sequence in a).
c) Modify the polynomial involved in creating f such that at least one power of ten becomes a member of the image of f.

Remarks:
a) Should be simple for those with a pass on MS221;
b) Not solvable with the standard MS221 / M208 tools but I'll expect 221+ folk will accept the challenge;
c) No hints as it is the bonus question;

Solvers are nominated for the gallery of excellence.

Update:
( 1/5-'10)
What sort of a person are you? Do you give it a try? Or don't you think it isn't worth the time? Then may I ask, why read my blog in the first place? Do you only like the " TMA-sort-of-exercices " which are easy to solve and get marked? Then... mathematics is not for you. Think about it. It concerns: -you-.
Anyway, I designed parts b) and c) of the exercise for myself. I haven't find the answer yet. It's more difficult than I thought. But I am learning... I'll solve it.l

1. u1 = 1 x 1
u2 = 1 x 2
u3 = 2 x 3
u4 = 3 x 5
u5 = 5 x 8
u6 = 8 x 13
u7 = 13 x 21

We recognise the 2 columns of factors as two instances of the Fibonacci sequence.

Giving:

u8 = Fib(8)Fib(9) = 714
un = Fib(n)Fib(n+1)

From here we can use the Binet formula (to get something horrible for the closed form)

2. That answer is worth 33% of the marks. A resit I am afraid. ;-)

The closed form is rather sexy.
1/10 (-2 (-1)^n-(Sqrt[5]-1) (1/2 (3-Sqrt[5]))^n+(1+Sqrt[5]) (1/2 (3+Sqrt[5]))^n)

phi = (1 + sqr(5))/2

Un = (1/5)((phi)^(2n+1) + (1-phi)^(2n+1) + (-1)^(n+1))

## 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.)