Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Exam of Discrete Mathematics which includes Recurrence Relation, Space Is Available, Answer, Number of Ways, Sum of Odd Integers, By Hand, Solution, Various Walks, Provided etc. Key important points are: General Solution, Recurrence Relation, Coefficients, Equations, Determine, Using, Answer, Calculate, Generating Functions, Binary Strings
Typology: Exams
1 / 7
This page cannot be seen from the preview
Don't miss anything!
(1 + x)n^ =
( n 0
)
( n 1
) x +
( n 2
) x^2 + · · · +
( n n
) xn
(1 − xn+1)
(1 − x)
= 1 + x + x^2 + x^3 + · · · + xn
1 (1 − x)
= 1 + x + x^2 + x^3 · · · =
∑^ ∞
i=
xi
1 (1 + x)n^
( −n 0
)
( −n 1
) x +
( −n 2
) x^2 + · · ·
∑^ ∞
i=
( −n i
) xi
( n + 1 − 1 i
) x + (−1)^2
( n + 2 − 1 2
) x^2 + · · ·
∑^ ∞ i=
(−1)i
( n + i − 1 i
) xi
1 (1 − x)n^
( −n 0
)
( −n 1
) (−x) +
( −n 2
) (−x)^2 + · · ·
∑^ ∞
i=
( −n i
) (−x)i
( n + 1 − 1 i
) (−x) + (−1)^2
( n + 2 − 1 2
) (−x)^2 + · · ·
∑^ ∞
i=
( n + i − 1 i
) xi
MACM 201 Test 2 March 8, 2006. 50 minutes Total marks: 55. Marks are indicated by ( ).
(1) (15) Consider the following recurrence relation;
an + 6an− 1 + 14an− 2 + 16an− 3 + 8an− 4 = 0, a 0 = − 1 , a 1 = 1, a 2 = 2, a 3 = 3
Write down the general solution and the equations that will determine the unknown coefficients, but do not solve for the coefficients.
(c) Calculate a 2 and a 3 using your answer from (b).
(3) (10) Solve the following recurrence relation using generating functions;
an+2 + 3an + 1 = n^3 + 2n, a 0 = 1, a 2 = 2
(5) (10) (a) Give an example of one loop-free graph with 5 vertices that has all of the following properties (indicate the features on your graph that satisfy these properties);
(i) There are no isolated vertices. (ii) There is exactly one (distinct) cycle of length 4. (iii) There is an edge such that if it is removed then the resulting graph has exactly two components.
(b) Give an example of a graph such that there is a trail between two distinct vertices that is not a path. (Recall that a trail is a walk where no edge is repeated, and a path is a walk where no vertex is repeated). Is it possible to have a path that is not a trail? Why or why not?
