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.