Problem 1. [20 points] Use mathematical induction to prove the following identities. (a) [5 points] For integers n 2 1, n n [(-1)12 = (-1)n > i=1 i= 1 Assume that En i = 2(+1 for all integers n 2 1. (b) [5 points] For integers n 2 2, n +1 = 2n (c) [5 points] Suppose that a1, a2, a3, . . . is a sequence defined as follows: a1 = 1, a2 = 8, ai = ai-1 + 20i_2 for all integers i 2 3. Prove that an = 3 . 27-1 + 2 . (-1)" for all integers n 2 1. (d) [5 points] Suppose that t1, t2, t3, ... is a sequence defined as follows: t1 = t2 = t3 = 1, ti = ti-1 + ti_2 + ti-3 for all integers i 2 4. Prove that tn