62. SUM OF NATURAL NUMBERS
 
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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


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