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in the last video we saw that if we were to take the sum from I equals 0 to n of I squared that this could be equal to and we did it in two videos it was a little bit hairy but we powered through this is equal to one-third n to the third power let me write a little bit cleaner than that and to the third power plus one-half n squared plus 1/6 1/6 then and that is absolutely true but you might have seen a different formula for this it doesn't look like that and so what I want to do is now manipulate this a little bit algebraically to come up with another formula that's typically seen for this sum so the first thing that we might realize is well let's what's what happens if we factor out a 1/6 n so this is going to be equal to let's factor out this right over here so that would be equal to 1/6 n times so let's see 1/3 divided by 1/6 that's the same thing as 1/3 times 6 so that's going to be equal to so this right over here is going to be equal to 2 N squared and you can verify it 1/6 times 2 is 2 6 or 1/3 n times n squared is n to the third power now 1/2 divided by 1/6 is the same thing as 1/2 times 6 which is 3 so this term right over here is going to be plus 3 3 times 3 times N and then finally you have this term right over here which is just going to be equal to 1 and now we're faced now we're faced with factoring this thing right over here now you could do this well let's do this by let's let's factor by grouping this and remember when you factor it by grouping you want to essentially break you want to break this into two numbers where the product of those two numbers the product of those two numbers is equal to the product of 2 times 1 and the obvious thing is well 2 times 1 is - if I break this into two and one if I break this three into two and one well two times one is definitely two times one so let me rewrite this I'm just factoring by grouping here nothing super fancy so if I rewrite this is 2n squared 2n squared plus plus n plus 2 n plus 1 plus 1 so this is all the stuff in this parenthesis right over here well this part right over here we can factor out an N so this is going to be equal let me rewrite the 1 6 2 n here so you know what I'm doing 1 6 then so if we factor out an N right over here we would have n times 2n n times 2n plus 1 2 n plus 1 plus 1 so that's what's in green right over here that's this part right over here and then what I have what I have here I could write this as plus 1 times 2n plus 1 1 times I'm gonna do that same color so 1 times 2n 2n plus 1 plus 1 and so now you can factor out a 2 n plus 1 and this thing simplifies to 1/6 n times 2n plus 1 I'm just factoring out a 2 n plus 1 I'm just factoring out a 2 n plus 1 and then you're going to have n plus if you take it if you factor a 2 n plus 1 of you out of each of these terms you're gonna have just an N and a 1 so 1/6 times n times 2n plus 1 2 n plus 1 times n plus 1 times n and plus 1 if this was a little bit confusing what I did here I encourage you to review the videos on factoring by grouping times n plus 1 and another way to rewrite this we could write this whole thing as being equal to and maybe I'll write in this green color that we started off with this is the same thing as this is the same thing and all right well we could I could write it as n times n plus 1 times 2 n plus 1 all of that all of that over 6 so this expression is equivalent to this expression and it's equivalent to this value right over here so whichever one you find to be a little bit easier to think about