Find the general form of the solution to a linear homogeneous recurrence with constant coefficients for which the characteristic roots are 1 with multiplicity 1, -2 with multiplicity 2 and 2 with multiplicity 3. The relation also has a non-homogeneous part which is a linear combination of n2n and (-2)n
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For the graph in Fig. Fig. 2 on the following page find a minimal colouring.
Let fn denote the nth Fibonacci number. Show by induction that every natural number is expressible as a sum
fn1 + fn2 + .......+ fnk
where ni - ni+1 > 1 for each i > 1.
Give examples of trivially true and vacuously true statements.
Which of the following statements are true and which are false? Give reasons for your answer.
i) ( ~p v q) and (p q) are logically equivalent.
ii) (m+n)! > m!+n! for m, n any positive integers.
iii) The number of distributions of m indistinguishable objects into n distinguishable containers is the same as the number of non-negative integral solutions of the equation x1+x2+....+xn = m.
iv) Sn, the number of subsets of a set with n elements, satisfies a second order linear homogeneous recurrence equation with constant coefficients.
v) K3,4 is Hamiltonian.
Give the order and the degree of the recurrence
a2 n+2 = a2n +2an+4
Is the recurrence homogeneous?
Is Petersen graph bipartite? Give reasons for your answer.
Which of the following sentences are statements? Give reasons for your answer.
i) Are the buses running today?
ii) What a pleasant weather!
iii) x2 +1 = 0 for a real value of x.
iv) Every odd number is a prime.