Consider the following recurrence relation:

$$T(n) = \begin{cases} \sqrt{n} T(\sqrt{n}) + n & \text{for } n \ge 1, \\ 1 & \text{for } n = 1. \end{cases}$$

Which one of the following options is CORRECT?

Consider functions Function_1 and Function_2 expressed in pseudocode as follows:

Function 1
   while n > 1 do
     for i = 1 to n do
       x = x + 1;
     end for
     n = n/2;
   end while

Function 2
  for i = 1 to 100 ∗ n do
    x = x + 1;
  end for

Let $$f_1(n)$$ and $$f_2(n)$$ denote the number of times the statement "$$x=x+1$$" is executed in Function_1 and Function_2, respectively.

Which of the following statements is/are TRUE?

Consider the following recurrence :

f(1) = 1;

f(2n) = 2f(n) $$-$$ 1, for n $$\ge$$ 1;

f(2n + 1) = 2f(n) + 1, for n $$\ge$$ 1;

Then, which of the following statements is/are TRUE?

For constants a ≥ 1 and b > 1, consider the following recurrence defined on the non-negative integers :

$$T\left( n \right) = a.T\left( {\frac{n}{b}} \right) + f\left( n \right)$$

Which one of the following options is correct about the recurrence T(n)?