Give a detailed proof that 314 is irrational. [Note: The number 314 can also be written as 141/3]. 2 For a positive integer n, a 1-near Skolem sequence of order n is a sequence with 2n 2 symbols such that (i) each integer j {2, . . . , n} occurs exactly twice, and (ii) the two occurrences of the integer j are exactly j positions apart. Some examples: 1-near Skolem seq., order n = 4: 3, 4, 2, 3, 2, 4 1-near Skolem seq., order n = 5: 3, 4, 5, 3, 2, 4, 2, 5 1-near Skolem seq., order n = 8: 3, 4, 8, 3, 7, 4, 6, 2, 5, 2, 8, 7, 6, 5. (a) Construct a 1-near Skolem sequence of order n = 9. (b) Prove that there is no 1-near Skolem sequence of order n = 31. 3. () Let x, y, z R. Prove that if 3x + y 60, then x + 2y 6z or z + 2x 6y. 4. ) Find all x Rsuch that 1 + |1 x|< 3. Prove your answer. [HINT: note that |y|< 3 if and only if 3 < y < 3; make cases on the sign of 1x.] 5. Prove that for each positive integer n, 14 + 25 + 36 + + n(n + 3) = n(n + 1)(2n + 10) 6 . 6. (4 points). Let a1 = 1 and for n >2 let an = ( n n+2)an1. Prove that an = 6 n2 + 3n + 2 for all n N