Use complete sentences. Your arguments should be clear and correct. You may use results that were proved in class or in the text without proving them again. B.1. For each positive integer in, let '71. TL '71. TE. '71. n) : n+k : n+0+n+1+n+2+m+n+a E.g.f(3)=+++g. Prove that n) is 8(a) B2. For each positive integer in, let T\" be the sum Tn = 1(1!) + 2(2!) + 3(3!) + + n(n!) = Zara). k=1 E.g. T3 = 1 + 4 + 18 = 23. Calculate a few values of TR to help you guess a simple formula for Tn. Then use induction to prove that your guess is correct for every positive integer n. (Recall that the factorial function is dened in Section 2.3.5, page 160. Observe that factorials satisfy the relation (71+ 1)! = (in. + 1)(n!) (1) 1 for every n E N. (E.g., 5! = 5 x (4!).) This is because (n+ 1) (n!) = (n+1) x [nx (n-1) x (n - 2) x . . . x 2 x 1] = (n +1)! We can also remark that because of the special definition 0! = 1, the equation (1) also holds for n = 0.) B.3. For which positive integers n is it true that n! 2 n2 ? Prove your answer. Part of your proof should use mathematical induction