PROBLEMS from volume 33 (1995) issue #2 of The Fibonacci Quarterly.

Copyright ◯^{c} 1995 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–784} Proposed by Herta Freitag, Roanoke, VA

Show that for all n,

α^{n−1}\sqrt{5} − L_{n−1}/α

is a Lucas number.

\underline{B–785} Proposed by Jane E. Friedman, University of San Diego, CA

Let a_{0} = a_{1} = 1 and let a_{n} = 5a_{n−1} − a_{n−2} for n ≥ 2. Prove that a_{n+1}^{2} + a_{n}^{2} + 3 is a multiple of a_{n}a_{n+1} for all n ≥ 1.

\underline{B–786} Proposed by Jayantibhai M. Patel, Bhavan’s R. A. College of Science, Gujarat State, India

If

F_{n+2k}^{2} = aF_{n+2}^{2} + bF_{n}^{2} + c(−1)^{n},

where a, b, and c depend only on k but not on n, find a, b, and c.

\underline{B–787} Proposed by H.-J. Seiffert, Berlin, Germany

For n ≥ 0 and k > 0, it is known that F_{kn}/F_{k} and P_{kn}/P_{k} are integers. Show that these two integers are congruent modulo R_{k} − L_{k}.

[Note: P_{n} and R_{n} = 2Q_{n} are the Pell and Pell-Lucas numbers, respectively, defined by P_{n+2} = 2P_{n+1} + P_{n}, P_{0} = 0, P_{1} = 1; and Q_{n+2} = 2Q_{n+1} + Q_{n}, Q_{0} = 1, Q_{1} = 1.] \underline{B–788} Proposed by Russell Jay Hendel, University of Louisville, Louisville, KY

(a) Let G_{n} = F_{n^{2}}. Prove that G_{n+1} ∼ L_{2n+1}G_{n}. [Note: f(n) ∼ g(n) means that f is asymptotic to g, that is, lim_{n→∞} f(n)/g(n) = 1.]

(b) Find the error term. More specifically, find a constant C, such that

G_{n+1} ∼ L_{2n+1}G_{n} + CG_{n−1}.

\underline{B–789} Proposed by Richard André-Jeannin, Longwy, France

The Lucas polynomials, L_{n}(x), are defined by L_{0} = 2, L_{1} = x, and L_{n} = xL_{n−1} + L_{n−2}, for n ≥ 2.

Find a differential equation satisfied by L_{n}^{(k)}, the kth derivative of L_{n}(x), where k is a nonnegative integer.