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Real smart with students swayed a quick request what was about this identity right if if you want to sum of a bunch of numbers right. Whats the sum for that okay. Thats a very famous thing right its been around gauss apparently.
I think it was like eight or something he figured this out. I think the story something like he was in third grade. They were sitting around the teacher basically sat there and said you know what i need i got to do stuff so.
Im gonna like these kids add up numbers from 1 to 100 ok. That should keep them busy so apparently baby gauss right. He sat there went.
Why do i have to do this according to him it was obvious that one in a hundred adds up to 101. And then he said well you know what if you go up 1 and you go down. 1.
This sum is still the same okay. Then you realize well you got 50 pairs of these. So then he thought ok.
The son must be 50 50. And the story is he walks up you know takes writes it on his little chalkboard tosses on the teachers desk. Ok.
So all good i guess right ok. But so were gonna do that that idea is basically his idea and were just getting a little practice of this. And its a common idea like if you have some some a finite sum right you can manipulate that sum in different ways to get a nice pretty formula for it here.
Its difficult because these guys are all increasing by 1 you can think of different ways to do that you can visualize. It as dots things like that there are a lot of like neat ways of seeing it. But i think maybe one cheesy way to practice is exactly what gauss said these two guys and those guys they would all add up to same sum right so i think what i want to do is maybe.
Write. It. Backwards.
So. This. Is m.
N. Minus. 1.
Plus. N. Minus.
2. Plus. 1.
Degree. Its the same sum. We just went from 1 up to n.
And now were going from 1 up to n. And we just broke them in a different order. Okay the nice thing about this.
Though is if you add the two of them of course. N. Plus.
1. Is n. Plus.
1. Right. And you can see.
This. Term. 2 plus.
N. Minus. 1.
Well. Its still n. Plus.
1. I mean. This is basically what you did right and then you look across at the end.
Its still n plus. 1. And of course.
Thats twice. Youre some everybody agree so. Now.
The only thing is left left is mellow each each guy adds up to n. Plus. 1.
How many terms. You have you have. 1 2.
3. 4. 5.
6. 7. 8.
9. N. Of them right.
So. Yeah. N.
Of these guys. So. I.
Guess. The total sum has got to be n. Times n.
Plus. One thats equal to twice heres some right. So.
The sum of course has got to be n times n. Plus. 1.
Over. 2. Okay.
So heres maybe another way. We can do the same thing. So we want to add up different guys right get some sort of sum.
Okay you can maybe visualize. It a lot of different ways to experiment and play with this stuff. Lets take a concrete example lets say n is equal to.
5. Ok so now we have something looks like this so theres 5 right there ok. Then if i wanted to add 4 4.
Would be like this that would be the number below him right. Because it would be 1 plus. 2 plus 3 plus.
4. Plus. 5.
Then i would have three then two and then one thats not bad thats actually counting this sum right. Its not as pretty though because if i have something like a square that would be really pretty then itd be easy for me to count right because then you can just do. 5 times.
5. Fact lets play on play with that okay so maybe we extended. Whats missing in the square.
I think we have 1 1. 2. 1.
2. 3 1. 2.
3. 4. So do you guys agree with this it looks like if you add this total up.
Were going to get something pretty that looks like this ok. So what does that mean that means that if we have a sum that equals s. And then we have a son thats one step back right and we add them up we should get this 5 by 5 grid.
Which is n squared. So the number of dots. Here is definitely n.
Squared. Right. Ok you have some sum over there.
Which is so we have a sum s of these guys based on n. Ok. Then you have a sum which is the same sum right sing.
What so you had were shy of just one guy here so n. But agree so so buddy see that this guy here really is exactly the same sum youre just missing that row of 5 and that 5 is n so im going to do that so s plus s. Minus n.
Has got to be n squared ok no big deal but then this guy is twice the sum n is equal to n. Squared. Now.
Its just algebra. So we add the both sides twice n is equal to n. Squared plus n.
Divided by 2 you get this guy. If you want it to look familiar factor. Out that n.
Thats n plus 1. So factor that guy out like we said. Just a little bit of algebra.
Then that turns out to be what n times. N. Plus.
1. Over. 2.
Okay totally another way to see it .
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