Find with proof a formula for tn of the form tn an bn

Question

Putnam TNG - Sequences and Convergence

1: Let d be a real number. For each integer m ≥ 0, define a sequence {a_m(j)}, j = 0, 1, 2, ... by the conditions a_m(0) = d/2^m, a^m(j + 1) = (a^m(j))² + 2a_m(j) for j ≥ 0. Evaluate lim_n→∞ a_n(n).

2: Let (a_n) be a sequence of positive reals such that, for all n, a_n ≤ a_2n + a_2n+1. Prove that _n=1∑^∞a_n diverges.

3: Let T₀ = 2, T₁ = 3, T₂ = 6 and for n ≥ 3,

T_n = (n + 4)T_n-1 - 4nT_n-2 + (4n - 8)T_n-3.

The first few terms are

2, 3, 6, 14, 40, 152, 784, 5168, 40576, 363392.

Find, with proof, a formula for T_n of the form T_n = A_n + B_n, where (A_n) and (B_n) are well-known sequences.

4: Is √2 the limit of a sequence of numbers of the form n^1/3 - m^1/3 (n, m = 0, 1, 2, ...)?

5: Given a sequence (x_n) such that lim_n→∞(x_n - x_n-2) = 0, prove that

lim_n→∞ (x_n - x_n-1/n) = 0.

6: Let A = {(x, y): 0 ≤ x, y ≤ 1}. For (x, y) ∈ A, let

S(x, y) = ∑_{1/2 ≤ m/n ≤ 2}x^myⁿ,

where the sum ranges over all pairs (m, n) of positive integers satisfying the indicated inequalities. Evaluate lim_{(x,y)→(1,1),(x,y)∈A}(1 - xy²)(1 - x²y)S(x, y).

7: Assume that (a_n)_n≥1 is an increasing sequence of positive real numbers such that lim_n→∞ a_n/n = 0. Must there exist infinitely many positive integers n such that

a_n-i + a_n+i < 2a_n for i = 1, 2, ... , n - 1?