## Special Series

In this page special series we are going to see formulas to find the sum of the special series.Here you can find four formulas.The first formula is the sum of natural numbers,second is sum of squares, third formula is for sum of cubes and fourth is for sum of odd numbers.

(i)    Sum of first n natural numbers

n (n + 1) / 2

(ii)    Sum of squares

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

(iii) Sum of cubes
[ n (n + 1) / 2 ]2

Example 1:

Find the sum of first 75 natural numbers.

Solution:

1 + 2 + 3 + 4 + ................. + 75

= 75 ( 75 + 1 )/2

= (75 x 76) / 2

= 75 x 38

= 2850

Therefore the sum of first 75 natural number is 2850.

Example 2:

Find the sum of 15 + 16 + 17 + .............. + 80

Solution:

= ( 1 + 2 + 3 +........+ 80 ) - ( 1 + 2 + 3 + ..........+ 14)

= [80 (80+1)/2] - [14 (14+1)]/2]

= [80x 81]/2 - [14 x 15]/2

= (40 x 81) - (7x15)

= 3240 - 105

= 3135

Example 3:

Find the sum of 1 + 4 + 9 + ............. + 1600

Solution:

The given series can be written in the form

12 + 22 + 32+............ +402

=  40 (40+1) (2x40 + 1) / 6

=  (40 x 41 x 81) / 6

= 20 x 41 x 27

=  22140

Example 4:

Find the sum of the following series

113 + 12 3 + ............... +k3 where k = 50
= (13 + 23 + ..........+50 3) - (13 + 23 + ..........+ 10 3)
= [(50 x 51) / 2]2 - [(10 x 11)/2] 2
= 1625625 - 3025
= 1622600

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: