PROBLEMS from volume 30 (1992) issue #3 of The Fibonacci Quarterly.

Copyright ◯^{c} 1992 by The Fibonacci Association.

                         Inquiries to: Fibonacci@MathPro.com

ELEMENTARY PROBLEMS AND SOLUTIONS Edited by Stanley Rabinowitz

Please send all material for ELEMENTARY PROBLEMS AND SOLUTIONS to Dr. STAN-LEY RABINOWITZ; 12 VINE BROOK RD; WESTFORD, MA 01886-4212 USA. Correspondence may also be sent to the problem editor by electronic mail to Fibonacci@MathPro.com on Internet. All correspondence will be acknowledged.

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BASIC FORMULAS

The Fibonacci numbers F_{n} and the Lucas numbers L_{n} satisfy

F_{n+2} = F_{n+1} + F_{n}, F_{0} = 0, F_{1} = 1;
L_{n+2} = L_{n+1} + L_{n}, L_{0} = 2, L_{1} = 1.

Also, α = (1 + \sqrt{5})/2, β = (1 − \sqrt{5})/2, F_{n} = (α^{n} − β^{n})/\sqrt{5}, and L_{n} = α^{n} + β^{n}.

PROBLEMS PROPOSED IN THIS ISSUE

\underline{B–718} Proposed by Herta T. Freitag, Roanoke, VA Prove that [(F_{n} + L_{n})α + (F_{n−1} + L_{n−1})] /2 is a power of the golden ratio, α.

\underline{B–719} Proposed by Herta T. Freitag, Roanoke, VA

Dedicated to Dr. A. P. Hillman Let P_{n} be the nth Pell number (defined by P_{0} = 0, P_{1} = 1 and P_{n+2} = 2P_{n+1} + P_{n} for

n ≥ 0). Let a be an odd integer. Show how to factor P_{n+a}^{2} + P_{n}^{2} into a product of Pell numbers.

How should this problem be modified if a is even?

\underline{B–720} Proposed by Piero Filipponi, Fond. U. Bordoni, Rome, Italy

Dedicated to Dr. A. P. Hillman

Find a closed form expression for

S_{n} = ∑_{h+k=2n} F_{h}F_{k}

where the sum is taken over all pairs of positive integers (h, k) such that h + k = 2n and h ≤ k.

\underline{B–721} Proposed by Russell Jay Hendel, Dowling College, Oakdale, NY

Dedicated to Dr. A. P. Hillman Brittany is going to ascend an m step staircase. At any time she is just as likely to stride

up one step as two steps. For a positive integer k, find the probability that she ascends the whole staircase in k strides.

\underline{B-722} Proposed by H.-J. Seiffert, Berlin, Germany

Dedicated to Dr. A. P. Hillman Define the Fibonacci polynomials by F_{0}(x) = 0, F_{1}(x) = 1, and F_{n}(x) = xF_{n−1}(x)+F_{n−2}(x),

for n ≥ 2. Show that for all nonnegative integers n,

∫_{0}^{∞} \frac{dx}{(x^{2} + 1)F_{2n+1}(2x)} = \frac{π}{4n + 2}.

\underline{B–723} Proposed by Bruce Dearden and Jerry Metzger, University of North Dakota, Grand Forks, ND

(a) Show that for n ≡ 2 (mod 4),

F_{n+1}(F_{n}^{2} + F_{n} − 1) divides F_{n}^{n}(F_{n}^{2} + F_{n+1}) − 1.

(b) What is the analog of (a) for n ≡ 0 (mod 4)?