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Evaluating series using the formula for the sum of n squares

Video transcript

let's try to find the sum of this write or the sum or let's try to evaluate this expression right over here so we're evaluating what this what this sum turns out to be now there's a bunch of ways to do this you could literally just do it by brute force you could see well what is this equal when N equals 1 when N equals 2 all the way to N equals 7 and that would be completely legitimate but I view this as an opportunity to look at some properties of Sigma notation so let's split this out so the first thing that you might say is well look if I were to sum up all of these if I were to if I were when N equals 1 and then N equals 2 all the way to N equals 7 it's reasonable that this is going to be the same thing as the sum of the sum of 3 N squared for men equals 1 to 7 plus plus plus the sum from N equals 1 to 7 of 2 N squared of 2 n I should say that right over there plus plus the sum plus the sum from N equals 1 to 7 of 4 and if this if you find this a little bit confusing I encourage you to expand both of these things out and realize that when you rearrange the terms that you will get these two things I'm not doing a rigorous proof here but hopefully if you were to expand this out you would see that this is not an unreasonable thing to claim right over here now out of all of these this last this last piece is pretty easy to evaluate when N equals 1 this thing is equal to 4 when N equals 2 this thing is equal to 4 when N equals 3 this thing is equal to 4 so you're essentially going to take 4 you're going to take 7 fours and add them together so this is essentially just going to evaluate to 7 times 4 or 28 now let's look at this piece right over here now we once again we could just do it by brute force 2 times 1 is 2 plus 2 times 2 is 4 so you're going to essentially do the first 7 multiples of 2 is one way that you could think about this or if you if we were to expand it out actually let me expand it out this is going to be 2 plus 4 plus 2 plus 4 plus 6 all the way all the way to 1 this is 7 all the way to 14 you could factor out a 2 and so this is going to become 2 times 1 plus 2 plus 3 all the way to 7 and so you can rewrite this piece right over here as the sum as 2 times the sum so we're essentially just factoring out the 2 2 times this sum which is the sum from N equals 1 to 7 of n so this is this piece we still have this 28 that we have to add so we have this 28 and we draw the parentheses so you don't think that the 28 is part of this right over here and now we can do the same thing with this 3 times n so we're taking from N equals 1 to 7 of 3 and squared doing the same exact thing over as we just did in magenta this is going to be this is going to be equal to 3 times the sum from N equals 1 to 7 of N squared we're essentially factoring out the 3 we're factoring out the 2 N squared and once again we could put parentheses just to clarify things now at this point there are formulas to evaluate to evaluate each of these things there's a formula to evaluate this thing right over here there's a formula to evaluate this thing over here and you can look them up and actually I'll give you the formulas if in case you're curious this formula one expression of this formula is that this would is going to be n to the third over 3 plus N squared over 2 plus n over 6 that's one formula for that and one formula for this piece right over here going from N equals 1 to 7 or sorry let me make it clear this this n is actually what your is what your terminal kind of value should be so this should be 7 to the third power over 3 so it's not this n I was just mindlessly using the formula 7 to the third over 3 plus 7 squared over 2 plus 7 over 6 so that's this sum and this sum you could view it as the average of the first and the last term so the first term is 1 the last term is 7 to take their average and then multiply it times the number of terms you have so times you have seven you have seven terms so what is this middle one going to evaluate to well one one times one two and of course we have this two out front this green is just this part right over here so you have two times this and over here you have three you have three times this business right over here so if we evaluate this one two times let's see 1 plus 7 is 8 divided by 2 is 4 4 times 2 is 8 times 7 it's 56 so that becomes 56 now this let's see this is actually well we could evaluate this if we want and I guess we could we could take out a calculator if we want to figure out 7/3 5 actually let's just do that just a save time here so let's calculate so we have 7 to the third power divided by 3 plus 7 squared divided by 2 plus 7 divided by 6 gives us a drumroll of 140 so this is going to be equal to this is equal to 3 times 140 let me do it in that color 3 times 140 plus 56 plus 28 that's 28 and since we get our calculator out let's just use it so let's see 140 times 3 this is 3 times 140 is 420 of course plus 56 56 plus 28 we deserve a drum roll now gets us to 504 so this sum this sum right over here is equal to 504 let's see in multiple ways you can do it but it's nice to know that there are these there's these kind of ways to break down the problem and there are these formulas now I encourage you I encourage you to look at the formulas to see how you proved how this is actually derived and proved I'm not a big fan of just saying oh there's a formula for this you just apply it you know the formula here is that is whatever this terminal value is to the third power for 3 plus that squared over 2 plus that squared or plus that over 6 I encourage you to look up on our on our site on Khan Academy the formula for the sum of n squares and will tell you where this is derived from and also the slop the formula for the sum of an arithmetic series and it'll tell you where this is derived from