answer:PGM

answer:The first thing to notice is that for large n, we have F(n) = n - c (in this case, n > 7^21), meaning that only values of n less than or equal to b truly matter when computing S(a,b,c).Furthermore, since we have a recursive loop of 4 a's in F(n) = F(a + F(a + F(a + F(a + n)))), we want to find the number of n's where n % (4 * a) range from 0 to b. From these n's, we can then calculate F(n), even for large numbers, due to the cyclical nature of F(n). We store the values of F(n) for 0 <= n < 4*a in an array, denoted arr[].Next, observe that the series S(a,b,c) is equivalent to b*a times the sum of arr[i] divided by a (the average of arr), plus the sum the remaining elements of arr that are within b (i.e., for n such that n % (4 * a) <= b).To further clarify the process, consider the example of S(50, 2000, 40):- We see that for n<2000, there are 2000/200 = 10 sets of F(i), using i range 0- 199.- Therefore, sum(F(i)) for 0 <= i < 2000 equals to 10*sum(F(i)) for 0 <= i < 200.- After 10*200 points, there are additional 0 points less than 2000. So, for these points, sum(F(i)) adds to the previous total sum to be the final result.We can use this process to compute S(a,b,c) for large numbers, and the code snippet below does exactly that (taking the last 9 figures):```pythondef S(a, b, c): MOD = 109 M = 4 * a # repetition of F(n) arr = [0] * M for n in range(M - 1, -1, -1): # top down calculation if n > b: arr[n % M] = n - c else: arr[n % M] = arr[(a + arr[(a + arr[(a + arr[(a + n) % M]) % M]) % M) % M] return ((b // M * sum(arr)) % MOD + sum(arr[i] for i in range(b % M + 1))) % MODprint(S(217, 721, 127))```The top-down calculation ensures that if a number contributes to future calculations, its value has been computed already. At last, we calculate the circular sum as described earlier and return the result.

answer:Sure, I can augment the code to produce and print the first ten prime numbers based on the `is_prime` function. Have a look at the Python script below:```pythondef is_prime(n): if n == 1: return False for i in range(2, n): if n % i == 0: return False return Truecount = 0num = 2while count < 10: if is_prime(num): print(num) count += 1 num += 1```In this script, I've used a while loop to keep incrementing `num` from 2, checking each time whether it's prime. When it is, it's printed out, and `count` is incremented. The loop stops after finding the first ten primes.

answer:Those `#define` directives at the top of the file form a construction known as an "include guard". In C and C++, when you `#include` a file, the preprocessor essentially copies and pastes the contents of the included file into the including file. If a file gets included multiple times, this can cause problems, as it would result in multiple definitions of the same things.To prevent this, most C and C++ header files use include guards. This is the structure:```c#ifndef SOME_UNIQUE_IDENTIFIER#define SOME_UNIQUE_IDENTIFIER// the contents of the header file go here#endif // SOME_UNIQUE_IDENTIFIER```The `#ifndef` directive checks whether the given identifier has been `#define`d. If it hasn't, it `#define`s it and includes the rest of the file. If it has, it skips over the rest of the file. This means that, no matter how many times the file is `#include`d, its contents only end up in the including file once.Now, for your case, it seems your environment is automatically adding this structure even to .c files, not just headers (.h files). This is generally not necessary or useful, since .c files don't usually get `#include`d. It won't harm anything, but you should be free to remove it if you wish.

answer:You'll need to adjust the table and column names based on your database schema, but the general query would look something like this:```sqlSELECT *FROM EmployeesWHERE DATE_PART('year', AGE(Current_Date, Hire_Date)) > 10 AND Annual_Salary < 50000;```This statement assumes you have a table called "Employees" with columns "Hire_Date" and "Annual_Salary". It uses the AGE function to calculate the number of years between the current date and the "Hire_Date", then filters for the employees that have been with the company for over 10 years. It also filters for employees with "Annual_Salary" less than 50,000. Again please replace the table name and column names to match your actual table in the database.

answer:The R script supplied isn't complete, and hence it's hard to pinpoint any syntax errors. However, there is clearly an issue with the command as it is. In R, `<-` is the assignment operator used to assign a value (right side) to a variable (left side). The `c()` is a function to combine the elements into a vector. If I correctly interpret your intention, you try to assign a vector containing the numbers 1, 2 and 3 to the variable `x`. The problem here is that your command isn't finished. You opened a bracket without closing it which causes a syntax error. Below is the corrected version:```Rx <- c(1, 2, 3)```This script will run without any errors and result in creating a vector `x` with elements 1, 2, and 3.