PROBLEMS from volume 32 (1994) issue #2 of The Fibonacci Quarterly.

Copyright ◯^{c} 1994 by The Fibonacci Association.

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ELEMENTARY PROBLEMS AND SOLUTIONS Edited by Stanley Rabinowitz

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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–760} Proposed by Russell Euler, Northwest Missouri State University, Maryville, MO

Prove that F_{n+1}^{2} ≥ F_{2n} for all n ≥ 0.

\underline{B–761} Proposed by Richard André-Jeannin, Longwy, France Evaluate the determinants

| \begin{matrix}L_{0} & L_{1} & L_{2} & L_{3} & L_{4} \\ L_{1} & L_{0} & L_{1} & L_{2} & L_{3} \\ L_{2} & L_{1} & L_{0} & L_{1} & L_{2} \\ L_{3} & L_{2} & L_{1} & L_{0} & L_{1} \\ L_{4} & L_{3} & L_{2} & L_{1} & L_{0} \end{matrix} |  and  | \begin{matrix}L_{0}^{2} & L_{1}^{2} & L_{2}^{2} & L_{3}^{2} & L_{4}^{2} \\ L_{1}^{2} & L_{0}^{2} & L_{1}^{2} & L_{2}^{2} & L_{3}^{2} \\ L_{2}^{2} & L_{1}^{2} & L_{0}^{2} & L_{1}^{2} & L_{2}^{2} \\ L_{3}^{2} & L_{2}^{2} & L_{1}^{2} & L_{0}^{2} & L_{1}^{2} \\ L_{4}^{2} & L_{3}^{2} & L_{2}^{2} & L_{1}^{2} & L_{0}^{2} \end{matrix} | .

\underline{B–762} Proposed by Larry Taylor, Rego Park, NY Let n be an integer.

(a) Generalize the numbers (2, 2, 2) to form three three-term arithmetic progressions of integral multiples of Fibonacci and/or Lucas numbers with common differences 3F_{n}, 5F_{n}, and 3F_{n}.

(b) Generalize the numbers (4, 4, 4) to form two such arithmetic progressions with common differences F_{n} and F_{n}.

(c) Generalize the numbers (6, 6, 6) to form four such arithmetic progressions with common differences F_{n}, 5F_{n}, 7F_{n}, and F_{n}.

\underline{B–763} Proposed by Juan Pla, Paris France Let

A = ( \begin{matrix}e^{iπ/3} & \sqrt{2} \\ \sqrt{2} & e^{−iπ/3} \end{matrix} ) .

Express A^{n} in terms of Fibonacci and/or Lucas numbers.

\underline{B–764} Proposed by Mark Bowron, Tucson, AZ

Consider row n of Pascal’s triangle, where n is a fixed positive integer. Let S_{k} denote the sum of every fifth entry, beginning with the kth entry, \binom{n}{k}. If 0 ≤ i < j < 5, show that |S_{i} − S_{j}| is always a Fibonacci number.

For example, row 10 of Pascal’s triangle is 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1. Thus S_{0} = 1 + 252 + 1 = 254, S_{1} = 10 + 210 = 220, and 254 − 220 = 34 = F_{9}.

\underline{B–765} Proposed by Mohammad K. Azarian, University of Evansville, Evansville, IN

Let m and n be positive integers greater than 1, and let x = F_{mn}/(F_{m}F_{n}). What famous constant is represented by

[∑_{i=0}^{∞} (∑_{j=0}^{i} \frac{1}{j!}) (\frac{1}{x^{i}} − \frac{1}{x^{i+1}} )]^{x}?