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EXAMPLE 4.9 The Factorial Numbers
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The factorial numbers 0!, 1!, 2!, 3!, are defined recursively by the equations 0! = 1 n! = n ( n 1 )
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n n! 0 1 1 1 2 2 3 6 4 24 5 120 6 720
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For example, letting n = 1 in the second equation yields 1! = 1((1 1)!) = 1(0!) = 1(1) = 1 Similarly, with n = 2: 2! = 2((2 1)!) = 2(1!) = 2(1) = 2 and with n = 3: 3! = 3((3 1)!) = 3(2!) = 3(2) = 6 The first seven factorial numbers are shown in the table at right. This program prints all the factorial numbers up to an input limit: int main() { long bound; cout << "Enter a positive integer: "; cin >> bound; cout << "Factorial numbers < " << bound << ":\n1, 1"; long f=1, i=1; do { f *= ++i; cout << ", " << f; } while (f < bound); } Enter a positive integer: 1000000 Factorial numbers < 1000000: 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880 The do..while loop iterates until its control condition (f < bound) is false.
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4.4 THE for STATEMENT The syntax for the for statement is
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for (initialization; condition; update) statement; where initialization, condition, and update are optional expressions, and statement is any executable statement. The three-part (initialization; condition; update) controls the loop. The initialization expression is used to declare and/or initialize control variable(s) for the loop; it is evaluated first, before any iteration occurs. The condition
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expression is used to determine whether the loop should continue iterating; it is evaluated immediately after the initialization; if it is true, the statement is executed. The update expression is used to update the control variable(s); it is evaluated after the statement is executed. So the sequence of events that generate the iteration are: 1. evaluate the initialization expression; 2. if the value of the condition expression is false, terminate the loop; 3. execute the statement; 4. evaluate the update expression; 5. repeat steps 2 4.
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EXAMPLE 4.10 Using a for Loop to Compute a Sum of Consecutive Integers
This program has the same effect as the one in Example 4.1 on page 60: int main() { int n; cout << "Enter a positive integer: "; cin >> n; long sum=0; for (int i=1; i <= n; i++) sum += i; cout << "The sum of the first " << n << " integers is " << sum; } Here, the initialization expression is int i=1, the condition expression is i <= n, and the update expression is i++. Note that these same expressions are used in the programs in Example 4.1 on page 60, Example 4.4 on page 62, and Example 4.8 on page 64.
In Standard C++, when a loop control variable is declared within a for loop, as i is in Example 4.10, its scope is limited to that for loop. That means that it cannot be used outside that for loop. It also means that the same name can be used for different variables outside that for loop. EXAMPLE 4.11 Reusing for Loop Control Variable Names
This program has the same effect as the one in Example 4.1 on page 60: int main() { int n; cout << "Enter a positive integer: "; cin >> n; long sum=0; for (int i=1; i < n/2; i++) // the scope of this i is this loop sum += i; for (int i=n/2; i <= n; i++) // the scope of this i is this loop sum += i; cout << "The sum of the first " << n << " integers is " << sum << endl; } The two for loops in this program do the same computations as the single for loop in the program in Example 4.10. They simply split the job in two, doing the first n/2 accumulations in the first loop and the rest in the second. Each loop independently declares its own control variable i.
Warning: Most pre-Standard C++ compilers extend the scope of a for loop s control variable past the end of the loop.