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.
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How many numbers from 0 to 999(0 and 999 inclusive) are not divisible by 5 or 13?
Express 3x4+2x3 - 2x2+x in terms of [x]4, [x]3, [x]2 and [x].
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.
Let A be an 88 Boolean matrix (i.e. every entry is 0 or 1). If the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. Further, show that there are at least 4 such pairs of rows and columns.
Is the graph in Fig. 1 Eulerian? Is it Hamiltonian? Justify your answers.
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?
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.