PROBLEMS from volume 32 (1994) issue #1 of The Fibonacci Quarterly.

Copyright ◯^{c} 1994 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

The problems in this issue all involve Pell numbers. The Pell numbers P_{n} and their associated numbers Q_{n} satisfy

P_{n+2} = 2P_{n+1} + P_{n}, P_{0} = 0, P_{1} = 1;
Q_{n+2} = 2Q_{n+1} + Q_{n}, Q_{0} = 1, Q_{1} = 1.

If p = 1 + \sqrt{2} and q = 1 − \sqrt{2}, then P_{n} = (p^{n} − q^{n})/\sqrt{8}, and Q_{n} = (p^{n} + q^{n})/2. The Pell-Lucas numbers, R_{n}, are given by R_{n} = 2Q_{n}. For more information about Pell numbers, see Marjorie Bicknell, “A Primer on the Pell Sequences and Related Sequences.” Fibonacci Quarterly 13.4 (1975):345–349.

\underline{B–754} Proposed by Joseph J. Kostal, University of Illinois at Chicago, IL Find closed form expressions for

∑_{k=1}^{n} P_{k}  and  ∑_{k=1}^{n} Q_{k}.

\underline{B–755} Proposed by Russell Jay Hendel, Morris College, Sumter, SC Find all nonnegative integers m and n such that P_{n} = Q_{m}.

\underline{B–756} Proposed by the editor Find a formula expressing P_{n} in terms of Fibonacci and/or Lucas numbers.

\underline{B–757} Proposed by H.-J. Seiffert, Berlin, Germany Show that for n > 0,

(a)  P_{3n−1} ≡ F_{n+2} (mod 13),

(b)  P_{3n+1} ≡ (−1)^{⌊(n+1)/2⌋}F_{4n−1} (mod 7).

\underline{B–758} Proposed by Russell Euler, Northwest Missouri State University, Maryville, MO

Evaluate

∑_{k=0}^{∞} \frac{k2^{k}Q_{k}}{5^{k}} .

\underline{B–759} Proposed by H.-J. Seiffert, Berlin, Germany Show that for all positive integers k and all nonnegative integers n,

∑_{j=0}^{n} F_{k(j+1)}P_{k(n−j+1)} = \frac{F_{k}P_{k(n+2)} − P_{k}F_{k(n+2)}}{2Q_{k} − L_{k}} .