The University of Sydney School of Mathematics and Statistics
MATH2069 Assignment
MATH2069: Discrete Mathematics and Graph Theory Semester 1, 2018
This assignment is for students enrolled in MATH2069. It is due before 4pm on Thurs- day May 3 and must be submitted in an electronic form which is a scanned copy of your answers. The assignment solutions must be uploaded via Turnitin in the Learning Management System (Blackboard). It should include your name and SID. Printed/typed solutions are acceptable. You may quote any result that you need from lectures. All arguments and working must be shown.
1. (a)
Solve the recurrence relation
an =an−1 +8an−2 −12an−3 +25·(−3)n−2 +32n2 −64n, n3,
2. (a)
(5 marks)
Two sequences an and bn are related by the identities
bn = (−1)n(n+1)a0 +(−1)n−1na1 +···+(−1)2an−1 +an
which hold for all n 0. Find an expression for the generating function for the sequence bn in terms of the generating function for the sequence an.
wherea0 =130,a1 =215anda2 =260.
(b) Write down a closed formula for the generating function of the sequence an.
(b) Use your solution to part (a) to prove the identity α n α+2
n = (−1)k(k+1) n−k k=0
for all n 0, where α is a complex number.
Copyright ⃝c 2018 The University of Sydney
(5 marks)