Average of number series

Average and sum of natural numbers series

1 .Sum of all first n natural numbers is equal to

Sum=n(n+1)/2

and we know that 

Average= sum/number

so average of first n natural numbers is 

Average=n(n+1)/2n=(n+1)/2



2. Sum of first n odd numbers

=n²

Average of first n odd numbers=n

3. Sum of first n even numbers

=n(n+1)

Average of first n even numbers=n+1

4. Sum of the squares of first n natural numbers

=n(n+1)(2n+1)/6

Average =(n+1)(2n+1)/6


5. Sum of cubes of first n natural numbers

=[n(n+1)/2]²

Average= n(n+1)²/4

6.The average of n consecutive number is always the middle number where n is odd. but when n is even then average will be average of two middle terms.

eg average of 2,4,6,8,10. 
as n=5 so average will be 6.

average of 2,4,6,8,10,12
as n=6 average will be (6+8)/2=7


7.The sum of squares of first n consecutive even numbers 
=2n(n+1)(2n+1)/3

Average=2(n+1)(2n+1)/3

8. The sum of squares of first n consecutive odd numbers 
=n(n+1)(2n+1)/3

Average=(n+1)(2n+1)/3

9. If the average of n consecutive numbers is x then the difference between the smallest and the largest number is given by 2(n-1).


10. Average of a series having common difference 2 is given by

(first term+last term)/2.