Let P(n) be the statement that a postage of n cents can be formed using just 4-cent stamps and 7-cent stamps. The Induction and Recursion parts of this exercise outline a strong induction proof that P(n) is true for n > 18. (a) Show that P(18), P(19), and P(20) are true, which com- pletes the base case.
(b) What is the inductive hypothesis?
(c) What do you need to prove in the inductive step?
(d) Complete the inductive step for k > 20.

Robots with telescoping arms are sometimes used to perform tasks (e.g., welding or placing screws) where access may be difficult for other robotic types. during a test run, a robot arm is programmed to extend according to the relationship r = 3 + 0.5cos(4θ) and the arm rotates according to the relationship θ=−π4t2+πt , where r is in feet, θ is in radians, and t is in seconds. use a computer program to plot the path of tip a in x and y coordinates for 0 ≤ t ≤ 4s.

Jasper and samantha are in a robotics competition. the guidelines state that the robots should be able to move a 10-gram weight at least 2 meters and turn in a circle. jasper and samantha have already built the robot. which step of the design process should they follow next to decide whether their robot meets the minimum criteria for the competition?

1. ask the user how many questions are in the quiz. 2. ask the user to enter the key (that is, the correct answers). there should be one answer for each question in the quiz, and each answer should be an integer. e.g., 34 7 13 100 81 3 9 10 321 12 might be the key for a 10-question quiz. you will need to store the key in an array. 3. ask the user to enter the answers for the quiz to be graded. there needs to be one answer for each question. note that these answers do not need to be stored; each answer can simply be compared to the key as it is entered. 4. when the user has entered all of the answers to be graded, print the number correct and the percent correct. 5. add a loop so that the user can grade any number of quizzes with a single key. after the results have been printed for each quiz, ask "grade another quiz? (y/n)." note: you only have one array (the key). you are not creating a new key for each set of quiz answers.