1) Prove that if loga x = y, then logb x = (loga x)(log, a). Plot logan for 1 n 100 when a = 2, e and 10. 2) Prove that (a) (b) (c) i=n(n+1)/2 = n(n+1)(2n+1)/6 =1 i=0 x = (xk+-1)/(x-1) if x > 1 = (1 - xk+)/(1-x) if x < 1 Hint: Induction - Discrete Math (10-points) (15-points) 3) Consider two balanced dices with each of the six faces marked 1 to 6. In one single throw of these two dice, list all the possible outcomes. Compute the average value of the outcomes. (10-points) 4) Compute the size of the problem that can be solved on a machine that takes 10-12 sec/op in one day when T(n) = 4 n, 10 n, 2n. (10-points)