1) Consider a1, a2, a3, ... be sequence defined by a1 = 3/2 and an = 5an-1 -1 for n >= 2. Write first 6 terms of sequence. Guess the formula for an, and prove that the guess is correct.
2) Assume a0, a1, a2, ... is sequence such that a0 = a1 = 1 and for n >= 1, an+1 = n (an + an-1).
i): Determine a2, a3, a4, and a5.
ii): Guess the formula for an, valid for n >= 0, and utilize mathematical induction to prove that the guess is correct.
3) Let sequence defined by a1 = 1, an+1 = (n + 1)^2 - an for n >= 1. Determine first six terms. Guess the general formula for an and prove that answer is right.