Give tight asymptotic upper bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for sufficiently small n. Please give me a step by step answers, thanks.

Let a0 = 1, an = 6an−1 + 3n be the recurrence relation of algorithm A and T(1) = 1, T(n) = 2T(n/5)+n be the recurrence relation of algorithm B. Find the solution for both relations using the Characteristic Polynomial method. Without proof, find Big-Oh for both algorithms. Which algorithm is faster? Justify your answer using Big-Oh notation.

The possible solution to the recurrence relation T(n)= 2T(n/4) + 1     a θ(n)   b θ(n2)   c θ(nlgn)   d θ(√n)

Algorithms

Solve the first-order linear recurrence relation: Sn+1 = Sn + 2, with S0=1. You may use the general solution given on P.342.

Express the solution in big-O terms for the following recurrence relation: T(n) = 9*T(n/3) + n^3; T(1)=1; Answer: T(n) = O(......) .

It is an algorithm problem.

Prove by induction that, if T(n) ≤T(5n/6) + O(n), then T(n) = O(n). Assume the base case isconstant, i.e., that T(n) = Θ(1) for all n≤cfor some constant c. Then, prove this result againusing the DC Recurrence Theorem

What is time complexity of T(n) = 3T(n/2) + C, using the recurrence equations? Please help me

Let ap = 1, an = an-1 + 3 be the recurrence relation of algorithm A and T(1) = 1, T(n) = T(n/5)+n be the recurrence relation of algorithm B. 1. Find the solution for both relations. Use any method.