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.

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