Compulsory Exercises (Tuesday 27 October 2015)
Exercise 3.1 Show that . (You need to find the constants and for both the Big-O and Big- bound.)
Exercise 3.2 Find exact solutions to the following recurrence equations
- ,
- ,
Exercise 3.3 Find exact solutions to the following recurrence equations
- ,
- ,
Exercise 3.4 Consider the claim there is no largest prime number. (In other words, the set of prime numbers is infinite.) Complete the following proof for the claim.
We make the proof by contradiction, so we assume that there is a largest prime number which we call .
Let be one more than the product of all the primes from 2 to inclusive. In other words
where is the list of all primes from 2 to .
We can show that is not divisible by any prime because by the definition of .
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Exercise 3.5 Consider the Fibonacci sequence . Let denote the th number in this sequence. Give a recursive definition of .
Exercise 3.6 Use mathematical induction to show that for we have
where is the th number of the Fibonacci sequence.