62. SUM OF NATURAL NUMBERS | |||
---|---|---|---|

Home | Menu | Previous | Next |

Consider

The first **9** natural numbers.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

1 & 9 make 10; so can be replaced by 5 & 5

2 & 8 make 10; so can be replaced by 5 & 5

Same with the other symmetrically placed pairs.

This **5** is the middle number.

Average of first 9 natural numbers is also 5.

So, **average** is the middle number;

**Average** is half of **end-numbers;**

Also of symmetrically placed numbers

like 2 & 8, 3 & 7, ...

So, 1 + 2 + 3 + ... + 9

can become 5 + 5 + 5 + (9 times)

Their total is 9 x 5, which is 45.

So 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 is

1. **9** x 5

2. **9** x **Average**.

3. **9** x **Middle number**.

4. **9** x **half** of end numbers = 9x(9 + 1)/2

5. **9** x **Half** of Number next to last number = 9 x (10/2).

6. Half of [**9** x **Next Number**] or

6. [Half of **9**] x [**Next Number**] or

6. [**9**] x Half of [**Next Number**]

**Note**: One of the two consecutive numbers is always even, whose half may be taken.

EXAMPLES

1) 1 + 2 + 3 + 4 = 2 x 5 = 10

2) 1 + 2 + 3 + ... + 5 = 5 x 3 = 15

3) 1 + 2 + 3 + ... + 40 = 20 x 41 = 820

4) 1 + 2 + 3 + ... + 59 = 59 x 30 = 1770