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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