Number Tricks

Ms. Arroyo asked the class to see if they could find the sum of the first 50 odd numbers. As everyone settled down to their addition, Terry ran to her and said, "The sum is 2,500." Ms. Arroyo thought, "Lucky guess," and gave him the task of finding the sum of the first 75 odd numbers. Within 20 seconds, Terry was back with the correct answer of 5,625.

How does Terry find the sum so quickly?

 

Answer:

The following pattern holds: The sum is equal to n x n, when n is the number of consecutive odd numbers, starting with 1. For example, the sum of the first 3 odd numbers is equal to 3 x 3, or 9; the sum of the first 4 odd numbers is equal to 4 x 4, or 16; the sum of the first 5 odd numbers is equal to 5 x 5, or 25; and so on.