**Properties of Addition of Natural Numbers: **Sum of 1

^{st ‘}n’ Natural numbers = 1 + 2 + 3 + …. + n = $\frac{n\left( n+1 \right)}{2}$

Sum of squares of 1^{st }‘n’ natural number = 1^{2} + 2^{2} + 3^{2 }+ ….. + n^{2} = $\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}$

Sum of cubes of 1^{st }‘n’ Natural number = 1^{3 }+ 2^{3} + 3^{3} + ….. + n^{3} = ${{\left( \frac{n\left( n+1 \right)}{2} \right)}^{2}}$

Sum of 1^{st} ‘n’ Odd natural number = 1+ 3 + 5 + ….. + (2n – 1) = n^{2}

Sum of 1st ‘n’ even natural numbers = 2 + 4 + 6+ …….2n = n (n+1)

Sum of squares of 1^{st }‘n’ Odd natural number = 1^{2 }+ 3^{2} + 5^{2} + ….. + (2n – 1) ^{2} = $\frac{n\left( 2n-1 \right)\left( 2n+1 \right)}{3}$

Sum of squares of 1^{st} ‘n’ Even natural number = 2^{2} + 4^{2} + 6^{2} + ….. + (2n)^{2} = $\frac{2n\left( n+1 \right)\left( 2n+1 \right)}{3}$

Sum of cubes of 1^{st } ‘n’ Odd natural number = 1^{3 }+ 3^{3 }+ 5^{3} + ….. + (2n -1)^{3} = n^{2} (2n^{2} -1)

Sum of cubes of 1^{st } ‘n’ Even natural number = 2^{3} + 4^{3} + 6^{3} + ….. + (2n)^{3 }= 2 [n (n + 1)]^{2}

**Some examples for the above properties:**

*Find the sum of:*

*(A) First 30 natural numbers *

*(B) Squares of first 30 natural numbers *

*(C) Cubes of first 30 natural numbers *

*(D) First 30 odd natural numbers *

*(E) Squares of first odd 30 natural numbers *

*(F) Squares of first even 30 natural numbers *

*(G) Cubes of first odd 30 natural numbers *

*(H) Cubes of first even 30 natural numbers*