Problem 3. Let f(n) be a function of positive integer n. We know: We know: f(1) = f(2) = ... = f(1000) = 1 and for n > 1000 f(n) = 5n + f(n/1.01). Prove f(n) = O(n). Recall that x is the ceiling operator that returns the smallest integer at least x.

Problem 4. Let f(n) be a function of positive integer n. We know: f(1) = 1 f(n) = 10 + 2 f(n/8). Prove f(n) = O(n 1/3 )

Problem 3. Let f(n) be a function of positive integer n. We know: We know: f(1)=f(2)==f(1000)=1 and for n>1000 f(n)=5n+f(n/1.01). Prove f(n)=O(n). Recall that x is the ceiling operator that returns the smallest integer at least x. Problem 4. Let f(n) be a function of positive integer n. We know: f(1)f(n)=1=10+2f(n/8). Prove f(n)=O(n1/3)