Calculus, all content (2017 edition)
Sum of n squares (part 3)
The sum of the first n squares, 1 + 4 + 9 + 16 + ... + n², is given by the formula ⅙n(n+1)(2n+1). In this video we factor and rewrite the formula that we found in the previous video and obtain the common formula given above. Created by Sal Khan.
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- If the sum of an arithmetic sequence is an arithmetic series, then would the sum of a geometric sequence be a geometric series?(2 votes)
- Yes, you are correct sir!
- Does this sort of thing continue?
Would the sum of cubes end up being divided by 24 with a quartic up top?
Is it always a one degree higher polynomial divided by the original power factorial?
Or am I just hoping it stays this 'easy?'(1 vote)
- You could write it like that:
but there's a bit of cancelling you could do to make it:
However interestingly it can also be written as:
which is the square of the sum to n of the natural numbers(2 votes)
- wow ! i ´ve done this same strategy to the sum of the cubics and for elevated for four. I´m trying to generalize for a sum of terms elevated to a generic exponent, someone know how to do that ?(1 vote)
- If what I want to add instead of i^2 is 1/i^2 or even 1/1+i^2, what would be the correct formula?(1 vote)
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 1/3 n to the third power. Let me write it a little cleaner than that. n to the third power plus 1/2 n squared plus 1/6 n. And that is absolutely true, but you might have seen a different formula for this that 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, 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 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 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 with factoring this thing right over here. Now, you could to do this-- well, let's factor by grouping this. And remember, when you factor by grouping, you want to essentially break this into two numbers where 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 2. If I break this into 2 and 1, if I break this 3 into 2 and 1, well, 2 times 1 is definitely 2 times 1. So let me rewrite this. I'm just factoring by grouping here, nothing super fancy. So if I rewrite this as 2 n squared plus n plus 2n plus 1-- so this is all the stuff in this parentheses right over here-- well, this part right over here we can factor out an n. So this is going to be equal to-- let me just write the 1/6 n here so you know what I'm doing. So if we factor out an n right over here, we would have n times 2n plus 1. So that's what's in green right over here. That's this part right over here. And then what I have here, I could write this as plus 1 times 2n plus 1. Let me do it in that same color, so 1 times 2n plus 1. And so now, you can factor out a 2n plus 1. And this thing simplifies to 1/6 n times 2n plus 1. I'm just factoring out a 2n plus 1. And then if you factor a 2n plus 1 out of each of these terms, you're going to just an n and a 1. So 1/6 times n times 2n plus 1 times n plus 1. And if this was a little bit confusing what I did here, I encourage you to review the videos on factoring by grouping. And another way we can rewrite this, we could write this whole thing as being equal to-- and maybe I'll write it in this green color that we started off with. This is the same thing as-- and I'll write it as n times n plus 1 times 2n plus 1, 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.