Write a closed form expression terms of n and simplify your

Question

Complete Assignment. Solution to be given problems.

All working and graph should be included

1. Use Mathematical Induction to prove that, for all integers n ≥ 5,

4n + 1 < 2ⁿ.

2. Let _i=1Σⁿ a_i = n² - n, and a_o = 4.

(a) Write a closed form expression for _i=1Σ^n-1 a_i in terms of n and simplify your expression.

(b) Write a closed form expression for _i=0Σ^n-1 a_i in terms of n; simplify your expression.

(c) Show that if n ≡ 0 (mod 3), _i=0Σ^n-1 is divisible by 3.

(d) Show that if n ≡ 0 (mod 3), then a_n ≡ 1 (mod 3).

3. Let T = {3t| t ∈ Z} , Q = {5q | q  ∈ Z}, R = {6r | r ∈ Z}, S = {T, Q, R}.

(a) For each of the following statements either give a reason why it is true or give a counter-example to show it is false.

(1) Q ⊆ R
(ii) T ⊆ R
(iii) R ⊆ T
(iv) ø ∈ S
(v) ø ⊆ T

(b) Write down the set P(S).