1. Consider a sequence {an} defined by recurrence: a1 =1, and an = 2an-1+5 for n 2. Prove using induction that an = 3 2n - 5 for any n21. 2 subsets A and

8. Suppose ao, a1, a2, ... is a sequence such that do = a₁ = 1 and, for n ≥ 1, an+1 = n(an + an-1). (a) [BB] Find a2, a3, a4, and a5. (b) Guess a formula for an, valid for n ≥ 0, and use mathematical induction to prove that your guess is ollo correct.

Suppose that an satisfies a recurrence relation, and that an is an approximation to an with relative error En. It is known that no member of the exact sequence ao, a1, is zero, and also that |En+1| > 1.5|en|. If a N = 0, and a N-1, ãN-2,... are calculated by downwards recurrence, what is the smallest value for N such that 16 correct significant figures are guaranteed in ag? N =

Find the ordinary generating function for sequence defined by the recurrence: a0 = 2, an + 1 = 3an + 1 (n is greater than or equal to zero).