Consider the following recurrence, defined for n a power of 4 (for the time of some algorithm): T(n) = 3 if n = 1 2T(n/4) + 4n + 1 otherwise (a) Calculate T(16) by hand. Show your work. (b) Use the tree method to solve the recurrence exactly, assuming n is a power of 4. (c) Use the formula to calculate T(1). Show your work. (d) Use the formula to calculate T(4). Show your work. (e) Use the formula to calculate T(16). Show your work. (f) Use Mathematical Induction to prove that your formula is correct in general.

Fill in the following program so that it will correctly calculate the price of the orange juice the user is buying based on the buy one get one sale.#include //main functionint main() { int cartons; float price, total; //prompt user for input information printf("what is the cost of one container of oj in dollars? \n"); scanf(" [ select ] ["%d", "%c", "%f", "%lf"] ", & price); printf("how many containers are you buying? \n"); scanf(" [ select ] ["%d", "%c", "%f", "%lf"] ", & cartons); if ( [ select ] ["cartons / 2", "cartons % 1", "cartons % 2", "cartons % price", "cartons / price", "cartons / total"] [ select ] ["=", "==", "! =", "< =", "> =", "< "] 0) total = [ select ] ["price * cartons", "cartons * price / 2 + price", "(cartons / 2) * price", "cartons / (2.0 * price)", "(cartons / 2.0) * price + price", "((cartons / 2) * price) + price"] ; else total = ((cartons / 2) * price) + price; printf("the total cost is $%.2f.\n", total); return 0; }