To make short, below is described sum from 1 to 10.

First, imagine blocks defined as 1, 2, 3, - - - 8, 9, 10 (see Figure below).

Sum from 1 to 10 is obtained by counting the number of these blocks.

You may count it as "1+2+3+4+5+6+7+8+9+10," which would be difficult for you to do.

To alternately imagine, separate the right half of a block (blue) and turn it round by 180o to place it on its left half (white).

If seeing Figure above, you could understand that you can count it as (1+10)*5.

Thus, sum from 1 to 100 is (1+100)*50=5050

Japanese sites Mathematical Formulas| Kodawari House| Pinpoint StreetView| |