Adding a sequence of consecutive odd numbers

1. Choose a 2-digit odd number. Add all the odd numbers starting with one through this 2-digit number:
2. Add one to the 2-digit number.
3. Divide this sum by 2 (take half of it).
4. Square this number. This is the sum of all odd numbers from 1 through the 2-digit number chosen.

   Example:

1. If the 2-digit odd number selected is 35:
2. 35+1 = 36 (add 1).
3. 36/2 = 18 (divide by 2) or 1/2 x 36 = 18 (multiply by 1/2).
4. 18 x 18 = 324 (square 18): 18 x 18 = (20 - 2)(18) = (20 x 18) - (2 x 18) = 360 - 36 = 360 - 30 - 6 = 324.
5. So the sum of all the odd numbers from one through 35 is 324.

   See the pattern?

1. If the 2-digit odd number selected is 79:
2. 79+1 = 80 (add 1).
3. 80/2 = 40 (divide by 2) or 1/2 x 80 = 40 (multiply by 1/2).
4. 40 x 40 = 1600 (square 40).
5. So the sum of all the odd numbers from one through 79 is 1600.