## Special Series

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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

1

^{2} + 2

^{2} + 3

^{2}+............ +40

^{2}
= 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

11

^{3} + 12

^{3} + ............... +k

^{3} where k = 50

= (1

^{3} + 2

^{3} + ..........+50

^{3}) - (1

^{3} + 2

^{3} + ..........+ 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.”