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COMP9020 Answer each of the following five (5) questions: 1. A sequence is a function. What then, is the domain and co-domain of a sequence? 2. The Generalised Law of Distribution is as follows: c · n∑ k=m ak = n∑ k=m (c · ak) Explain in your own words why it is that this law is permissible. It might help in your answer if you imagine yourself explaining this law to someone for whom it is novel and surprising. 3. Where n = 3 and r = 2, compute the following:√( n× n r ) 4. Here is a recursively defined sequence a0, a1, a2 . . . for all integers k ≥ 2: ak = ak−1 + kak−2 + 1 (recurrence relation) (1) a0 = 1 and a1 = 2 (initial conditions) (2) Find a5, a6, and a7 Note: You must show your working. 5. Prove the following bymathematical induction. For the inductive step, do not use the method of direct proof. Instead, you must use proof by contradiction: For all integers n ≥ 1, it is the case that: 1 + 2 + 3 + . . .+ n = n(n+ 1) 2 Note: you must show all steps of the proof, justify each step, and identify when the explicit contradiction is reached. You must explain also why it is that the contradiction in question is in fact a contradiction. This is the end of your questions. There are no more questions. 1 Your instructions are as follows: Please submit your answers on Moodle under “Assignments - Problem Set 4”. You may either typeset your answers or do them with a pen/pencil and paper and upload a photograph. All normal file types are accepted. Each question is worth one (1) mark. Each problem set is worth five (5) marks. Each problem set is worth 5% of your final mark. It is your responsibility to check and ensure that the file that you have uploaded to Moodle (1) is the correct file, (2) is a readable file, and (3) has uploaded correctly/is not corrupted. Re-uploads are available before the due date. Re- uploads are not available after the due date. If a corrupted or incorrect file remains on Moodle after the due date for a problem set, then you will receive zero (0) marks for that problem set. Please contact your tutor in the first instance with all and any questions that you have concerning your problem set mark(s). This is the end of your instructions. There are no more instructions.