Ms. Hannify asked her class to see if they could find the sum of the first fifty odd numbers. As everyone settled down into their addition, T.J. ran to her and said, "The sum is 2500." Ms. Hannify thought, "Lucky guess," and gave him the task of finding the sum of the first 75 odd numbers. Within 20 seconds, T.J. was back with the correct answer of 5625.
How did T.J. complete his tasks so quickly?
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The sum of the terms of a finite arithmetic progression is S n = 2 ( a 1 + a n ) n , where a 1 is the first term, a n is the n-th term and n is the number of terms.
If we will sum up the first n odd numbers, the first term will be 1 and the n-th term will be 2 n − 1 . Then, we have:
S o d d = 2 ( 1 + 2 n − 1 ) n = 2 2 n 2 = n 2 .
We can conclude that the sum of the first n odd numbers is n 2