School of Mathematics and Statistics MAST30012 Discrete Mathematics, Semester 2 2020 Assignment 3 Student Name Student Number Tutor’s Name Practice Class Day/Time Submission deadline is 11.59pm Wednesday 21st October. This is a GradeScope assignment and should considered as a practise run for your end of semester exam which will be written and submitted in a similar way. • You need to print out this assignment and write your answers into the boxes below each question. It is recommend you work on a separate piece of paper then neatly copy your final answers into the boxes. • If extra space is required you may continue your answer on a separate page at the end of the assignment : – Write in the answer box that your answer is continued at the end of the assignment. – Write at the top of the extra page which question is being completed. • If you are unable to print out the assignment you may write on the pdf. • This assignment needs to be scanned or photographed then uploaded as a pdf. The first page must be this cover page, the next three pages are your answers in boxes and this is then followed by any extra pages of answers. • Full working must be shown in your solutions. • Marks will be deducted for incomplete working, insufficient justification or incorrect notation. Assignment continued on next page... Q1: Consider the recurrence relation an+3 = 2an+2 + an+1 − 2an, with a1 = 2, a2 = 5, a3 = 10. (a) Calculate an for n = 0 and for 4 ≤ n ≤ 6. (b) Prove (or disprove) that the generating function g(x) = ∞! n=0 anx n = 1 (1− x)(1 + x)(1− 2x) . Assignment continued on next page... (c) Use a partial fraction expansion of g(x) to find an exact expression for an. (d) There exists another sequence (bn)n≥0 such that for all n > 0, an = "n i=0 bi. Find the recurrence relation and initial values for bn. Assignment continued on next page... Q2: (a) Using only “algebraic” operations (eg. addition, multiplication, differentiation etc.) on the geometric series, h(x) = " n≥0 x n, find expressions for the generating functions of the counting sequences with terms i) an = n 3. ii) bn = n! k=0 k3. End of Assignment.
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