(a) Use mathematical induction to prove that 2n < n! for every integer n with n> 4. (Note that this inequality is false for n = 1,2 and 3.) (b) A culture of bacteria doubles each 3 hours. If there are 200 bacteria present at the beginning, how many bacteria will be after 24 hours? Show your work (c) Counting your two parents, four grandparents, eight great grandparents etc. how many direct ancestor? Use your Knowledge about sequences and series to prove and show your work. (d) For each of the following, say whether the series coverges or diverges. Show your work. n+3n-4 2n-4n+6 i. ii. 2n iii. (3/2) iv. n+ n-n (e) Show whether Ecoverges or diverges. Show your work (f) A ball dropped from a height of 50 cm. Each time it bounces, it returns to of its previous height. How far will the ball travel in the air before coming to rest? Use your knowledge about sequences and series topic to solve this question. See the following picture to help you. Show your work. b b3 b4 (g) Find the sums of the following series. Show your work 3n-1-1 n=1 6-1 4 i. ., [4] [4] [4] [4] [2] [2] [4]