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# Une démonstration par récurrence

## Transcription de la vidéo

let's say we have the simplest of arithmetic sequences and probably the simplest of sequences a 1 2 we're going to start at 1 and just increment by 1 1 2 3 and we're going to go all the way to N and what I want to think about is what is the sum of this sequence going to be in the sum of a sequence we already know we call a series so what is the sum and I'll just call it S sub n what is this going to be equal to 1 plus 2 Plus 3 plus going all the way all the way to N well we're going to do a little trick here where I'm going to rewrite this sum so I'll write it again is S sub n S sub n but now I'm just going to write it in reverse order I'm going to write it as n plus n minus 1 plus n minus 2 all the way to 1 and now I'm going to add these two equations so we know that S sub n is equal to this so we're adding the same thing to both sides of this equation up here so on the left hand side you're going to have S sub n plus S sub n is just 2 times S sub n and on the right hand side and this is where we start to see something kind of cool you have a 1 plus an n which is just going to be n plus 1 you have a 2 plus n minus 1 which is well 2 plus n minus 1 is going to be n plus one again plus n plus 1 you have a 3 plus n minus 2 well that's going to be n plus one again n plus 1 I think you see what is going on here and you're going to go all the way to this last this last pair I guess you could say it or call it this last these last two terms and we have an n plus 1 again plus n plus 1 so how many of these n plus ones do we have well we have n of them there were n of these terms in each of these in each of these equations 1 2 3 all the way to n so we can rewrite this thing as 2 times s of n is equal to you have n n plus 1 terms so we could write it as n times n plus one n times n plus one and now to solve for S sub n to solve our sum we can just divide both sides we can just divide both sides by two and so we are going to be left with the sum from 1 to N this arithmetic sequence we're just incrementing by 1 starting at 1 is going to be equal to n times n plus 1 over 2 and this is neat because now you can quickly find the sum let's say from 1 to 100 it'll be 100 times 101 over 2 so very quickly you can find the sums and what I'm curious about and what we'll explore in the next video is can we generalize this for any arithmetic sequence we started with a very simple one we started one we just incremented by 1 and it looks like it looks like this is so if I were to write it this way this is n times this is n times n plus 1 over 2 so this right over here this n this is the nth term in our sequence and this right over here this one was the first term in our sequence so at least in this case it looks like I took the average of the first term and the nth term so this right over here make it clear this is the average actually do it like this this right over here is the average of our of a sub 1 and a sub N and then I'm multiplying that times n and what I'm curious about is whether this is going to be true for any arithmetic sequence that the sum of it is going to be the average of the first and the last term times the number of terms