## Nifty Formulae for Natural Numbers

# Some properties of natural numbers :

**1•Sum of first n natural numbers : 1 + 2 + 3 +……+ n = n ( n +1)/2**

**2• Sum of squares of first n natural numbers : 1² + 2² + 3² +……+ n² = n ( n +1) ( 2n +1) /6**

**3• Sum of cubes of first n natural numbers : 1³ + 2³ + 3³ +……+ n³ = [ n ( n +1)/2]²**

**4• Sum of first n odd numbers: n²**

**5• Sum of first n even numbers: n² + n**

**6• Sum of squares of 1st n Odd natural number : 1² + 3² + 5² +……+ ( 2n -1)² = n ( 2n -1) ( 2n +1)/3**

**7• Sum of squares of 1st n even natural numbers : 2² + 4² + 6² +…..+ (2n)² = 2n ( n +1) ( 2n +1) /3**

**8• Sum of cubes of 1st n odd natural numbers : 1³ + 2³ +3³ +……+ ( 2n -1)³ = n² ( 2n² -1)**

**9• Sum of cubes if first n even natural numbers : 2³ + 4³ + 6³ +…….+ 2n³ = 2[n ( n+1)] ²**

## Properties of Perfect Squares

**1• Digital sum of a perfect square is 1, 4, 7 or 9.**

**2• All perfect squares are of the form 3n or 3a**

**n + 1.**

**3• All perfect squares are of the form 4n or 4n + 1.**

**4•If unit**** digit of a perfect square is 1 then ten’s digit has to be Even.**

**5• If unit digit of a number is 2 then it can’t be a perfect square.**

**6•If unit digit of a number is 3 then it can’t be a perfect square.**

**7•If unit digit of a perfect square is 4 then its ten’s digit has to be Even.**

**8•If units digit of a perfect square is 5 then ten’s digit has to be 2.**

**9•If units digit of a perfect square is 6 then ten’s digit has to be odd.**

**10•If unit digit of a number is 7 & 8 then it can’t be a perfect square.**

**11•If unit**** digit of a perfect square is 9 then ten’s digit has to be Even.**

**12•If units digit of a perfect square is 0 then ten’s digit has to be 0.**

**More Gyan on Numbers **

Divisibility Rule of a Number (Divisibility Test)

Divisors of a Number ( Number of Divisors of a number)

Divisors of a Number-2: ( Number system – Number of Divisors)

Do you know we have a forum where you can get all your quant doubts answered?

http://tathagat.mba/forums/

Thank you sir. Please post some more shortcuts like this .