By Gillespie L. J.

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For this purpose it is useful to introduce first some definitions. 9. We define sum of any two sequences F and G as the sequence H whose element Hk is the sum of the corresponding elements of the two sequences: Hk = Fk + Gk . We will write: H = F + G. 10. We define product of a sequence F by a complex number α, the sequence W whose generic element is obtained by multiplying the corresponding element of F by α : Wk = αFk . We will write W = αF . 11. Two sequences F and G are said to be linearly dependent if there exist two constants α and β, not both zero, such that αF + βG is the zero sequence.

An−1 are given constants with a0 = 0. 6) and a term of the type ak m , where m is the multiplicity of the root 1 of the characteristic polynomial (m = 0 if 1 is not a root) and a is a constant to be determined by substitution. 19. Let us consider the difference equation Xk+2 − 2Xk+1 + Xk = 1. 13), has a characteristic equation with double root 1. We look for a solution of the given equation in the form Yk = ak 2 where a is a real constant to be determined. 13), we obtain: a (k + 2)2 + −2 a (k + 1)2 + ak 2 = 1; by matching the coefficients of the powers of k we obtain a = 1/2.

1. A pie is cut with straight cuts in such a way that every two cuts intersect in one and only one point, while three different cuts cannot intersect at the same point. Write a difference equation that describes recursively the number Nk of portions after k cuts. Exploit the result to answer the question: what is the maximum number of slices (possibly different from each other) in which you can divide a round cake by k straight cuts? 2. 8, verify that if R ≤ rS0 it is impossible to repay the debt!

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