PROBLEMS from volume 31 (1993) issue #1 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.

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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–730} Proposed by Herta Freitag, Roanoke, VA For n ≥ 0, express the larger root of

x^{2} − L_{n}x + (−1)^{n} = 0

in terms of α, the larger root of

x^{2} − x − |(−1)^{n}| = 0.

\underline{B–731} Proposed by H.-J. Seiffert, Berlin, Germany Evaluate the determinant:

| \begin{matrix}F_{0} & F_{1} & F_{2} & F_{3} & F_{4} \\ F_{1} & F_{0} & F_{1} & F_{2} & F_{3} \\ F_{2} & F_{1} & F_{0} & F_{1} & F_{2} \\ F_{3} & F_{2} & F_{1} & F_{0} & F_{1} \\ F_{4} & F_{3} & F_{2} & F_{1} & F_{0} \end{matrix} | .

Generalize.

\underline{B–732} Proposed by Richard André-Jeannin, Longwy, France

Dedicated to Dr. A. P. Hillman

Let (w_{n}) be any sequence of integers that satisfies the recurrence

w_{n} = pw_{n−1} − qw_{n−2}

where p and q are odd integers. Prove that, for all n,

w_{n+6} ≡ w_{n} (mod 4).

\underline{B–733} Proposed by Piero Filipponi, Rome, Italy

Write down the Pell sequence, defined by P_{0} = 0, P_{1} = 1, and P_{n+2} = 2P_{n+1} + P_{n} for n ≥ 0. Form a difference triangle by writing down the successive differences in rows below it. For example,

0  1  2  5  12  29  70  169 …
1  1  3  7  17  41  99
0  2  4  10  24  58
2  2  6  14  34
0  4  8  20
4  4  12
0  8

8

...

Identify the pattern that emerges down the left side and prove that this pattern continues.

\underline{B–734} Proposed by Paul S. Bruckman, Edmonds, WA If r is a positive integer, prove that

L_{5^{r}} ≡ L_{5^{r−1}} (mod 5^{r}).

\underline{B–735} Proposed by Curtis Cooper and Robert E. Kennedy, Central Missouri State University, Warrensburg, MO

Let the sequence (y_{n}) be defined by the recurrence

y_{n+1} = 8y_{n} + 22y_{n−1} − 190y_{n−2} + 28y_{n−3} + 987y_{n−4} − 700y_{n−5} − 1652y_{n−6} + 1652y_{n−7}

+ 700y_{n−8} − 987y_{n−9} − 28y_{n−10} + 190y_{n−11} − 22y_{n−12} − 8y_{n−13} + y_{n−14}

for n ≥ 15 with initial conditions given by the table:

\begin{matrix}n & | & y_{n} \\ 1 & | & 1 \\ 2 & | & 1 \\ 3 & | & 25 \\ 4 & | & 121 \\ 5 & | & 1296 \\ 6 & | & 9025 \\ 7 & | & 78961 \\ 8 & | & 609961 \\ 9 & | & 5040025 \\ 10 & | & 40144896 \\ 11 & | & 326199721 \\ 12 & | & 2621952025 \\ 13 & | & 21199651201 \\ 14 & | & 170859049201 \\ 15 & | & 1379450250000 \end{matrix}

Prove that y_{n} is a perfect square for all positive integers n.