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: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”