please explain the each answer, don't just write the answers.

Problem 1: Give the exact and asymptotic formula for the number f(n) of letters "A" printed by Algo- rithm PRINTAs below. Your solution must consist of the following steps: (a) First express f(n) using a summation notation (b) Next, give a closed-form formula for f(n). (c) Finally, give the asymptotic value of the number of A's (using the e-notation.) Include justification for each step. Note: If you need any summation formulas for this problem, you are allowed to look them up, and do not need to prove. Algorithm PRINTAS (n: integer) for it 1 to 2n + 1 do for jt 1 to (i + 3) do print("A") for i 1 to (2n + 1)2 do for j = 1 to i +3 do print("A") Problem 2: (a) Use mathematical induction to prove that 3" >n2" for n > 0. (Note: dealing with the base case may require some thought.) (b) Consider a sequence defined recursively as Bo = 1, B = 2, and B = BR-1 +33,-2 for n > 2. Prove that B = (2.4") and B = 12(2.3"). Hint: First, prove by induction that B . 2.3" and B. 5 2.4" for all n > 0. Problem 3: Give the asymptotic values of the following functions, using the e-notation: (a) (3 +5n -10)(4n- n) (b) 3+ 2n vn + 1/(log?n) (e) 7n log n + In log"n + 1506 (d) 15n + n2.2 + nalogn Justify your answers