Vol. 4 No 1 - Pi Mu Epsilon

16Then, if C = p - q6, D = p - qy,(7)It is not the purpose of this paper to investigate at any lengththe properties which one may derive by consideration of the matrices- D6ngiven above. Let is suffice to say, however, that many identitiesJn = c' result from such a study, and that one may, by this approach, avoid manyy- 6 '"messy" inductive proofs. In general, one would follow roughly thesame procedure as in Basin and Hoggatt [?I.and(9)Finally, we remark that it is possible to generate the sequence[S ] -- and, by (ll), the sequence (J] as well -- by the followingscneme: we may easily show that the fraction Sn+2/Sn is the n-thconvergent of the continued fractionConsider the matrixREFERENCESBy induction we have that1. L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea,1952.2. Charles R. Wall, Sums and Differences of Generalized FibonacciNumbers, to be published in Fibonacci Quarterly.3. A. F. Horadam, A Generalized Fibonacci Sequence, Amer. Math. Monthly,68 (l96l), 455-59.4. Stephen Fisk, Problem B-10, * Fibonacci Quarterly, l(19631, NO. 2,p. 85.We may easily verify, sincethatThus we may generate the sequence (J 1 by evaluating powers of the matrixs.It is obvious from (10) thatwhich, for Fibonacci numbers, is the basis for a famous geometricaldeception.5 S. L. Basin and V. E. Hoqqatt, Jr., A Primer on the FibonacciSequence, Part 11, Fibonacci Quarterly, 1 (l963), No. 2,pp. 61-68.RESEARCH PROBLEMS.tThis is a new section that will be devoted to suggestions of topickand problems for Undergraduate Research Programs. Address all correspondenceto the Editor.Proposed by M. S. W I N . Determine an "efficient" computer algorithmfor determining the center and radius of the smallest circle which coversa given set of points in a plane. Also, consider extensions to coveringwith equilateral triangles, squares, ellipses (say of minimum area,minimum major axis, . . . ) . . . , and to higher dimensions.