Explain an algorithm that computes 2^n using Θ(log n) multiplications. Justify why your algorithm uses Θ(log n) multiplications. a. Consider the case where n is a power of 2, i. e., n = 2^k . Use Θ(k) multiplications to obtain 2n . Hint: Note that 2^(2^k) = (2^(2^k−1) )^2 .
b. Consider the case where n = 2^k + 2^(k−1) . Use Θ(k) multiplications to obtain 2^n . Hint: Note that 2^(2^k+2^(k−1)) = 2^(2^k) × 2^(2^(k−1)) .
c. Generalize the ideas in parts (a) and (b) to compute 2n using Θ(log n) multiplications for any positive integer n.

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Write a pseudocode solution for each of these problems. 1. design a while loop that lets that user enter a number. the number should be multiplied by 10, and the result stored in a variable named product. the loop should iterate as long as product contains a value less than 100. 2. design a do-while loop that asks the user to enter two numbers. the numbers should be added and the sum displayed. the loop should ask the user whether he or she wishes to perform the operation again. if so, the loop should repeat; otherwise it should terminate. 3. design a for loop that displays the following set of numbers: 0, 10, 20, 30, 40, 50 100. 4. design a nested loop that displays 10 rows of # characters. there should be 15 # characters in each row. 5. convert this for loop to a while loop. declare integer count for count = 1 to 50 display count end for 6. find the error in the following pseudocode. declare boolean finished = false declare integer value, cube while not finished display “enter a value to be cubed.” input value; set cube = value ^ 3 display value, “ cubed is “, cube end while