PROBLEMS from volume 31 (1993) issue #4 of The Fibonacci Quarterly.

Copyright ◯^{c} 1993 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.

Each solution should be on a separate sheet (or sheets) and must be received within six months of publication of the problem. Solutions typed in the format used below will be given preference. Proposers of problems should normally include solutions. Proposers should inform us of the history of the problem, if it is not original. A problem should not be submitted elsewhere while it is under consideration for publication in this column.

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–748} Proposed by Herta T. Freitag, Roanoke, VA

Let u_{k} = F_{kn}/F_{n} for some fixed positive integer n. Find a recurrence satisfied by the sequence (u_{k}).

\underline{B–749} Proposed by Richard André-Jeannin, Longwy, France For n a positive integer, define the polynomial P_{n}(x) by

P_{n}(x) = x^{n+2} − x^{n+1} − F_{n}x − F_{n−1}.

Find the quotient and remainder when P_{n}(x) is divided by x^{2} − x − 1.

\underline{B–750} Proposed by Seung-Jin Bang, Albany, CA Find a linear transformation T: ℝ^{2} → ℝ^{2} such that T (F_{n}, L_{n}) = (F_{n+1}, L_{n+1}).

\underline{B–751} Proposed by Jayantibhai M. Patel, Bhavan’s R. A. College of Science, Gujarat State, India

Prove that 6L_{n+3}L_{3n+4} + 7 and 6L_{n}L_{3n+5} − 7 are divisible by 25.

\underline{B–752} Proposed by Richard André-Jeannin, Longwy, France

Consider the sequences (U_{n}) and (V_{n}) defined by the recurrences U_{n} = P U_{n−1} − QU_{n−2}, n ≥ 2, with U_{0} = 0, U_{1} = 1 and V_{n} = P V_{n−1} − QV_{n−2}, n ≥ 2, with V_{0} = 2, V_{1} = P, where P and Q are real numbers with P > 0 and Δ = P^{2} − 4Q > 0. Show that for n ≥ 0, U_{n+1} ≥ (P/2)U_{n} and V_{n+1} ≥ (P/2)V_{n}.

\underline{B–753} Proposed by Jayantibhai M. Patel, Bhavan’s R. A. College of Science, Gujarat State, India

Prove that for all positive integers n,

| \begin{matrix}F_{n−1}^{3} & F_{n}^{3} & F_{n+1}^{3} & F_{n+2}^{3} \\ F_{n}^{3} & F_{n+1}^{3} & F_{n+2}^{3} & F_{n+3}^{3} \\ F_{n+1}^{3} & F_{n+2}^{3} & F_{n+3}^{3} & F_{n+4}^{3} \\ F_{n+2}^{3} & F_{n+3}^{3} & F_{n+4}^{3} & F_{n+5}^{3} \end{matrix} | = 36.