As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

Please follow this blog

Search this blog

Tuesday, May 4, 2010

Exercise - comments

A reader ( Paddy ) surprised me with an excellent answer to the exercise I posted. The series I posted was in fact the running total of the squares of the Fibonacci numbers. Let me put that in a table.

$ \begin{matrix}
n & F(n) & (F(n))^2  & \Sigma\\
1 & 1 & 1 & 1\\
2 & 1 & 1 & 2\\
3 & 2 & 4 & 6\\
4 & 3 & 9 & 15\\
5 & 5 & 25 & 40\\
6 & 8 & 64 & 104\\
7 & 13 & 169 & 273
\end{matrix} $

Paddy's answer...

$ \begin{matrix}
n & F(n) & F(n) * F(n+1)\\
1 & 1 & 1 \\
2 & 1 & 2 \\
3 & 2 & 6 \\
4 & 3 & 15 \\
5 & 5 & 40 \\
6 & 8 & 104 \\
7 & 13 & 273
\end{matrix} $

So $ \sum_{k=1}^{n} F_n^{2} = F_n * F_{n+1} $
An interesting conjecture to prove formally.
Posted by at 8:09:00 PM
Labels: ,

1 comment:

  1. Webmaster@BoroughmuirWednesday, May 5, 2010 at 7:49:00 PM GMT+2

    [Sorry about the styling]
    If we look at Fn * Fn+1
    Fn+1 = Fn + Fn-1
    So
    Fn * (Fn + Fn-1)
    = Fn^2 + Fn * Fn-1
    Fn = Fn-1 + Fn-2 so we have
    Fn^2 + Fn-1 * (Fn-1 + Fn-2)
    = Fn^2 + Fn-1^2 + Fn-1 * Fn-2
    etc, etc.
    Works?

    ReplyDelete

Subscribe to: Post Comments (Atom)

Popular Posts

  • Video Lectures on Number Theory
    Among lectures on Calculus I,II and III, ( Introduction to ) Linear Algebra and ( Introduction to ) Differential Equations from the UCCS ( ...
  • Sum of a list in Mathematica
    Problem: We want to calculate the sum of the elements of a list of numbers. Suppose this list is named l and has been assigned the value {1,...
  • Ramanujan's magic square formula
    Today I started to read the Ramanujan biography ( The e-book version, of course. ) The book looks promising. What was it like to communicate...
  • Video lectures Abstract Algebra
    I found a set of video lectures on Abstract Algebra. MATH E-222 Abstract Algebra - http://www.extension.harvard.edu/openlearning/math222/ E...
  • Ramanujan's magic square formula - Revisited
    Ramanujan's genius (r) was discovered by Hardy (l) At a very young age Ramanujan designed the following formula for a 3 by 3 magic sq...

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.)