Assigned Questions (1) Let an be the sequence defined recursively as follows: a1 = 1 a2 = 1 an = an - 1 + 1 for n > 3. 2 an - 2 Prove that an E [1, 2] for all n. (2) Recall that Di = n(n+ 1) i= 1 2 (a) Use the above formula (not induction) to find and prove a formula for 2 + 4 + .. . + 2n. Express your final answer as a simplified fraction involving n. (b) Use induction to prove your formula in (a). (c) Use (a) to find and prove a formula for 1+ 3 + 5 + .. . + (2n -1). Express your final answer as a simplified fraction involving n. (d) Use induction to prove your formula in (c). (3) Let n, m E N with m i. Express your final answer as a =m simplified fraction involving n, m. (Hint: You should not need induction!) (4) For a natural number n, we define n! = 1 . 2 . 3 . ... . (n - 1) . n. For example: 2! = 1 . 2, 3! = 1 . 2 . 3 -6, 5! = 1 . 2 . 3 . 4 . 5 = 120. (a) Compute 6! and 7!. (In each case, simplify and express your final answer as a single integer.)MAT 102 - ASSIGNMENT # 4 DUE NOVEMBER 9 (b) Prove that if n, m E N, and n > m, then n! > m!. (c) Let at E R. Prove that for all n E N, if :1: Z n, then as\" Z n!. (5) Find all natural numbers n so that 6"+2 S 7\"_1. Prove your claim using induction. (6) Recall the Fibonacci sequence from Week 7 inclass activities. (a) Prove that Fn > 0 for all natural numbers n. (b) Prove that for all natural numbers n > 2 we have Fj