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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A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube?
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
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.
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.
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.
Show that any tree with exactly two vertices of degree 1 is a path.