Create a function that calculates the result of the arithmetic series

$\sum_{k = 0}^n (a + kd)$

where $a$ and $d$ are real numbers and $n$ is a positive integer. You can find a formula that gives you this result at Wikipedia.

You should also be able to determine this sum by using the following rules:

$\sum_{k = 0}^n (a_k + b_k) = \left( \sum_{k = 0}^n a_k \right) + \left( \sum_{k = 0}^n b_k \right)$
$\sum_{k = 0}^n ab_k = a \sum_{k = 0}^n b_k$
$\sum_{k = 0}^n 1 = n + 1$

Compare your formula you found for calculating the arithmetic series with the formula $(n + 1)(nd/2 + a)$. Are these formulas the same?

Your function declaration will be:

double arithmetic_series( double initial_value, double difference, unsigned int n );

You should not use any loops in your code.

The following function explicitly calculates the series, as opposed to using a formula. When this formula gives a different answer from yours, which do you suspect is more correct?

double arithmetic_series_explicit( double initial_value,
                                   double difference, unsigned int n );

double arithmetic_series_explicit( double initial_value,
                                   double difference, unsigned int n ) {
    double sum{initial_value};

    for ( unsigned int k{1}; k <= n; ++k ) {
        sum += initial_value + k*difference;
    }

    return sum;
}

Here is a program you can run:

#include <iostream>

double arithmetic_series_explicit( double initial_value,
                                   double difference, unsigned int n );
double arithmetic_series( double initial_value,
                          double difference, unsigned int n );
int main();

double arithmetic_series_explicit( double initial_value,
                                   double difference, unsigned int n ) {
    double sum{initial_value};

    for ( int k{1}; k <= n; ++k ) {
        sum += inial_value + k*difference;
    }

    return sum;
}

int main() {
    std::cout.precision( 16 );
    double a, d;
    unsigned int n;

    while ( true ) {
        std::cout << "Enter the initial value 'a': ";
        std::cin >> a;
        std::cout << "Enter the difference 'd': ";
        std::cin >> d;
        std::cout << "Enter the upper limit 'n': ";
        std::cin >> n;
    
        std::cout << arithmetic_series( a, d, n ) << std::endl;
        std::cout << arithmetic_series_explicit( a, d, n ) << std::endl;
    }

    return 0;
}