Here are some basic math formulas and identities that are commonly used in competitive programming:
- Arithmetic Progression (AP) formula:
- Sum of n terms of AP:
S = n/2(2a + (n-1)d) - nth term of AP:
an = a1 + (n-1)d
- Geometric Progression (GP) formula:
- Sum of n terms of GP:
S = a(1-r^n)/(1-r) - nth term of GP:
an = ar^(n-1)
- Pythagorean theorem:
a^2 + b^2 = c^2(where c is the hypotenuse of a right triangle, and a and b are the other two sides)
- Quadratic formula:
x = (-b + sqrt(b^2 - 4ac)) / 2a and x = (-b - sqrt(b^2 - 4ac)) / 2a(where a, b, and c are coefficients of a quadratic equation)
- Euler's formula:
e^(i*pi) + 1 = 0(where e is the base of the natural logarithm, i is the imaginary unit, and pi is the ratio of a circle's circumference to its diameter)
- Factorial formula:
n! = n * (n-1) * (n-2) * ... * 2 * 1
- Permutation formula:
nPr = n!/(n-r)!(where n and r are integers, and r is less than or equal to n)
- Combination formula:
nCr = n!/((n-r)! * r!)(where n and r are integers, and r is less than or equal to n)
- Binomial theorem:
(a+b)^n = a^n + nCa^(n-1)b + nC2a^(n-2)b^2 + ... + nb^n(where a and b are constants, and n is a positive integer)
- The sum of the first n natural numbers can be calculated using the following formula:
sum = n*(n+1)/2
where
n is the number of natural numbers to be added.For example, if we want to find the sum of the first 10 natural numbers, we can substitute
n = 10 in the formula:sum = 10*(10+1)/2 = 55
Therefore, the sum of the first 10 natural numbers is 55.