Odd numbers.

Logic Level 1

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?

He added 2450 to the number, then doubled it He doubled the number, then multiplied it by 10. He multiplied the amount of odd numbers by itself. He halved the number then multiplied it by 100.

2 solutions

Victor Paes Plinio
Jan 22, 2018

The sum of the terms of a finite arithmetic progression is $S_n=\frac{(a_1+a_n)n}{2}$ , 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 $2n-1$ . Then, we have:

$S_{odd}=\frac{(1+2n-1)n}{2}=\frac{2n^2}{2}=n^2$ .

We can conclude that the sum of the first $n$ odd numbers is $n^2$

I like to add the first and last and then square the result.

Marta Reece - 3 years, 4 months ago

The answer was to multiply the amount of odd numbers by itself, but that is also correct.

TJ Hannify - 3 years, 4 months ago

I did pick the answer, and understood it's applicability. But for actual on-the-fly calculations, I use mine.

Marta Reece - 3 years, 4 months ago

Edwin Gray
Sep 12, 2018

! + 3 + 5 + + (2n -1) = (n/2)( 1 + 2n -1) = n^2. Ed Gray

Odd numbers.

2 solutions

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