7 replies to this topic

[TheCube]

There was once a class in a far away land and other stupid crap you put in the beginning of a fairy tale. However the class was regular except for the teacher who we should call Mrs. Deaf because she screams so loud at her students that she made four of them lose their hearing. Everyone who knows her dread except for a younger student who we shall call Gauss because this is like a story I heard from ViHart with this same character.
Anyways, one day in Mrs. Deaf's math class before making Gauss deaf she assigned them a pop quiz with which she told them to find the sum of the first (insert number here) odd numbers, and all the questions were like this with the amount of odd numbers going above 100 for each one. However, Gauss finished it with ease handing it to Mrs. Deaf 3 minutes in. She told him to do it again because she knew it was all wrong and she didn't see any work of him adding the odd numbers. He told her "No, they are all right. If not, you can make me deaf."
"Deal" says the teacher thinking that she would win.
However, the paper was all right: Here's how it looked:

Find the sum for the first __ odd numbers:

112: 12544 350: 122500

222: 49284 500: 250000

    123: 15129


Furious at him, she broke her side of the deal and screamed at him, lying about how the paper was all wrong. The now deaf Gauss got up and as calmly as he came he went to his desk, waiting for Mrs. Deaf to make a look of sorry but she didn't.And the rest of the story doesn't really matter because the puzzle is already shown in front of us. The question is, how did Gauss sum up the odd numbers so quickly.

Spoiler for Hint

Try doing it yourself but with smaller numbers.

[BobbyGo]

Spoiler for Answer

The sum of the first x odd numbers == x²

Edited by BobbyGo, 29 May 2012 - 06:57 PM.

[phil1882]

Spoiler for more generally

n/2 *(t0 -tn) = s
as an example, for summing odds 7-27
10/2 * (7-27) = 100
as an interesting side note;
no prime can be expressed as the sum of consecutive odd numbers.
can you prove it?

[TheCube]

Spoiler for Answer

The sum of the first x odd numbers == x²

To be honest phil, this was the answer I was looking for.

[BobbyGo]

Spoiler for more generally

n/2 *(t0 -tn) = s
as an example, for summing odds 7-27
10/2 * (7-27) = 100
as an interesting side note;
no prime can be expressed as the sum of consecutive odd numbers.
can you prove it?

Spoiler for because

No prime can be expressed as the sum of consecutive odd numbers because...

the sum of two odd numbers, consecutive or otherwise*, would result in an even number.

Spoiler for *

If you want to get picky, i guess -3 + 5 = 2 would satisfy the otherwise* version.

Edited by BobbyGo, 30 May 2012 - 03:40 PM.

[TheCube]

Spoiler for because

No prime can be expressed as the sum of consecutive odd numbers because...

the sum of two odd numbers, consecutive or otherwise*, would result in an even number.

Spoiler for *

If you want to get picky, i guess -3 + 5 = 2 would satisfy the otherwise* version.

what about

Spoiler for spoiler

an odd amount of odd numbers will result in an even though. The thing with this though is that the median of the numbers is a factor of the sum besides 1 and itself so it makes it composite

[Rox53]

This is such an easy one. Ms. Deaf can't be much good at maths if she doesn't know the short cut!

Spoiler for spoiler for answer

The sum of the first n odd numbers is n squared 1+2 =4; 1+3+5=9; 1+3+5+7=16, etc

Good to have this forum back up and running!

[TheCube]

This is such an easy one. Ms. Deaf can't be much good at maths if she doesn't know the short cut!

Spoiler for spoiler for answer

The sum of the first n odd numbers is n squared 1+2 =4; 1+3+5=9; 1+3+5+7=16, etc

Good to have this forum back up and running!

Exactly,

I know right!

Oh, and Welcome to BrainDen!!!!