PROBLEM OF THE MONTH

October 2005




Congratulations to this month's winners &mdash Jerry McMahan Jr and Said Amghibech!
We received 5 solutions to this month's problem. Only two of them provide a complete proof that the limit exists.
Here is the solution by Jerry McMahan: [PDF]
Let a0, a1, a2, ... be the sequence given by the formula

an=3an-1 – an-2 for n > 1,
a0=a1=1

Prove that the sequence of ratios an/an-1 has a limit as n tends to infinity and find the limit.
Bonus question: Prove that the sequence defined above can be obtained by taking every other term of the Fibonacci sequence. Recall that the Fibonacci sequence f0, f1, f2, ...is given by the formula

fn=fn-1 + fn-2 for n > 1,
f0=f1=1

This month's prize will be awarded to the best explained, correct solution.


Submit your solution to the Mathematics Undergraduate Office (Math P-142) or electronically to problem@math.sunysb.edu by the due date. Acceptable electronic formats are: PDF, Postscript, DVI, (La)TeX, or just plain text. Please include your name and phone number, or preferably your email address.

Closing date: November 6th at 12 pm.