### The Fibonacci Sequence and Generalizations

Suppose that on January, 1st there are two newborn rabbits, 1 male and another female. What is the number associated with rabbits produced in a year if the following conditions hold: 1) each and every pair takes one month to get to maturity 2) each couple creates a mixed pair regarding rabbits on a monthly basis, from Feb on; and 3) no rabbits die throughout the 12 months. Assume that on Jan 1st there is 1 mixed pair of child rabbits.

At the beginning of Feb . there will still be one pair of rabbits considering that it takes a calendar month to allow them to become mature plus reproduce. In February one mixed pair of rabbits will be produced plus in March two sets will be produced, one from the original pair and one from the pair developed in February. After this design, in April, three sets will be produced, as well as in May five pairs. Considering that the 2× 2 matrix A new has only one distinct eigenvalue with multiplicity a couple of, it follows that A new is not diagonalizable. As a result, there is no general formula for Ak plus hence for yk.

Through the two illustrations above we are able to derive typically the following two conclusions concerning our matrix way for resolving second order recurrence relations: 1) If the attribute equation of A has two distinct solutions, then A is diagonalizable and we can derive a great explicit formula for yk. 2) If the characteristic equation of A offers below two distinct options, then the is not diagonalizable and our method for deriving yk can not be applied. A lot more formally, our conclusions will be consistent with the next more general theorem:

Theorem: Let function as the distinct solutions of the equation, where and m = 0. Then each solution of the geradlinig homogenous recurrence relation along with constant coefficients, where a0 = C0 and a2 = C1, is regarding the form for some constants A and B. 1 IV. Conclusion This paper discussed how linear algebra may be utilized to the analysis of linear recurrence relations, several which arise frequently inside applied and pure math. In particular, it highlighted the concepts of eigenvalues and eigenvectors and how helpful these are in explaining and understanding infinite sequences of integers.

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1 For the evidence of this specific theorem see Koshy, p. 143-4