| Tecnate | Last Updated: 2024-06-27 |
This is a reference document for some commonly used math concepts in programming, along with simple C++ examples for each.
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. General Formula
an = a1 + (n−1) ⋅ d
#include <iostream>
using namespace std;
int main() {
int a1 = 2; // first term
int d = 3; // common difference
int n = 5; // term to find
int an = a1 + (n - 1) * d;
cout << "The " << n << "th term of the arithmetic sequence is: " << an << endl;
return 0;
}
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
an = a1 ⋅ r(n−1)
#include <iostream>
#include <cmath>
using namespace std;
int main() {
double a1 = 3; // first term
double r = 2; // common ratio
int n = 4; // term to find
double an = a1 * pow(r, n - 1);
cout << "The " << n << "th term of the geometric sequence is: " << an << endl;
return 0;
}
Sn = (n/2)⋅[2a1 + (n−1) ⋅ d]
n terms#include <iostream>
using namespace std;
int main() {
int a1 = 1; // first term
int d = 1; // common difference
int n = 10; // number of terms
int Sn = (n / 2.0) * (2 * a1 + (n - 1) * d);
cout << "The sum of the first " << n << " terms of the arithmetic sequence is: " << Sn << endl;
return 0;
}
/
Sn = a1 ⋅ [(1 - rn) / (1 - r)]
n terms#include <iostream>
#include <cmath>
using namespace std;
int main() {
double a1 = 2; // first term
double r = 0.5; // common ratio
int n = 5; // number of terms
double Sn = a1 * (1 - pow(r, n)) / (1 - r);
cout << "The sum of the first " << n << " terms of the geometric sequence is: " << Sn << endl;
return 0;
}
The factorial of a non-negative integer n is the product of all positive integers less than or equal to n.
n! = n⋅(n−1)⋅(n−2)⋯3⋅2⋅1
#include <iostream>
using namespace std;
int main() {
int n = 5;
int factorial = 1;
for(int i = 1; i <= n; ++i) {
factorial *= i;
}
cout << "Factorial of " << n << " = " << factorial << endl;
return 0;
}
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1.
F(n) = F(n−1) + F(n−2)
#include <iostream>
using namespace std;
int fibonacci(int n) {
if(n <= 1) return n;
return fibonacci(n-1) + fibonacci(n-2);
}
int main() {
int n = 10; // nth Fibonacci number to find
cout << "The " << n << "th Fibonacci number is: " << fibonacci(n) << endl;
return 0;
}
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder.
#include <iostream>
using namespace std;
int gcd(int a, int b) {
if (b == 0)
return a;
return gcd(b, a % b);
}
int main() {
int a = 56, b = 98;
cout << "GCD is: " << gcd(a, b) << endl;
return 0;
}
The LCM of two integers is the smallest positive integer that is divisible by both numbers.
#include <iostream>
using namespace std;
int gcd(int a, int b) {
if (b == 0)
return a;
return gcd(b, a % b);
}
int lcm(int a, int b) {
return (a * b) / gcd(a, b);
}
int main() {
int a = 15, b = 20;
cout << "LCM is: " << lcm(a, b) << endl;
return 0;
}
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
A popular algorithm to find all primes less than or equal to a given integer n.
#include <iostream>
#include <vector>
using namespace std;
void sieveOfEratosthenes(int n) {
vector<bool> prime(n + 1, true);
prime[0] = prime[1] = false;
for (int p = 2; p * p <= n; p++) {
if (prime[p]) {
for (int i = p * p; i <= n; i += p)
prime[i] = false;
}
}
for (int p = 2; p <= n; p++) {
if (prime[p])
cout << p << " ";
}
cout << endl;
}
int main() {
int n = 50;
cout << "Prime numbers up to " << n << " are: ";
sieveOfEratosthenes(n);
return 0;
}
Modular arithmetic involves numbers wrapping around upon reaching a certain value, the modulus.
#include <iostream>
using namespace std;
int main() {
int a = 10, b = 25, n = 7;
int result = (a + b) % n;
cout << "(" << a << " + " << b << ") % " << n << " = " << result << endl;
return 0;
}
Exponentiation by squaring is a fast method for exponentiation that reduces the number of multiplications.
#include <iostream>
using namespace std;
long long power(long long x, int n) {
if (n == 0)
return 1;
long long half = power(x, n / 2);
if (n % 2 == 0)
return half * half;
else
return half * half * x;
}
int main() {
long long x = 2;
int n = 10;
cout << x << " raised to the power " << n << " is: " << power(x, n) << endl;
return 0;
}
#include <iostream>
#include <cmath>
using namespace std;
int main() {
double angle = 45; // in degrees
double radians = angle * M_PI / 180.0; // convert to radians
cout << "Sine of " << angle << " degrees is: " << sin(radians) << endl;
cout << "Cosine of " << angle << " degrees is: " << cos(radians) << endl;
return 0;
}
#include <iostream>
#include <cmath>
using namespace std;
int main() {
double number = 10.0;
cout << "Natural logarithm of " << number << " is: " << log(number) << endl;
cout << "Logarithm base 10 of " << number << " is: " << log10(number) << endl;
return 0;
}