Suppose we have a recursive sequence J1,. For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. For integer n ≥ 1, let P(n) be the predicate that i=1 fi = 2fn+2 - 3. Don't worry about whether this predicate "makes sense"; we haven't defined the fi so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first steps of the inductive sten are

(Program) Write a program that tests the effectiveness of the rand() library function. Start by initializing 10 counters, such as zerocount, onecount, twocount, and so forth, to 0. Then generate a large number of pseudorandom integers between 0 and 9. Each time 0 occurs, increment zerocount; when 1 occurs, increment onecount; and so on. Finally, display the number of 0s, 1s, 2s, and so on that occurred and the percentage of time they occurred.

The implementation of a queue in an array, as given in this chapter, uses the variable count to determine whether the queue is empty or full. You can also use the variable count to return the number of elements in the queue. On the other hand, class linkedQueueType does not use such a variable to keep track of the number of elements in the queue. Redefine the class linkedQueueType by adding the variable count to keep track of the number of elements in the queue. Modify the definitions of the functions addQueue and deleteQueue as necessary. Add the function queueCount to return the number of elements in the queue. Also, write a program to test various operations of the class you defined.

(Useful utility) Modify the program written for Exercise 3 to search for the first occurrence of a user-entered sequence of characters, and then replace this sequence, when it’s found in the string, with a second user-entered sequence. For example, if the entered string is Figure 4-4 illustrates the output of Program 4-2 and the user enters that 4- is to be replaced by 3-, the resulting string is Figure 3-4 illustrates the output of Program 4-2. (Only the first occurrence of the searched-for sequence has been changed.)