Integral Calculus (2017 edition)
- Summation notation
- Converting explicit series terms to summation notation
- Converting explicit series terms to summation notation (n ≥ 2)
- Summation notation intro
- Arithmetic series formula
- Worked example: arithmetic series (sigma notation)
- Worked example: arithmetic series (sum expression)
- Worked example: arithmetic series (recursive formula)
- Arithmetic series
Sal evaluates the arithmetic sum (-50)+(-44)+(-38)+...+2044. He does that by finding the number of terms and using the arithmetic series formula (a₁+aₙ)*n/2.
- [Voiceover] So we have the sum negative 50 plus negative 44, plus negative 38, all the way, we keep adding all the way up to 2,038, and then 2,044. So see if you can pause this video and evaluate this sum. So let's work through this together, and let's just think about what's going on. So the first term here is negative 50, and then we go to negative 44, so the second term is negative 50 plus 6, and then the third term, we add 6 again. Negative 44 plus 6 is negative 38, and we go all the way to here, we keep adding 6, and to go from 2,038 to 2,044, to get to that last term, we add 6 once again. And so each successive term is just 6 more than the term before it. So we are dealing right over here, this sum is an arithmetic series. It's a sum of an arithmetic sequence. Each term is 6 more, is a constant amount more than the term before that. So we know how to take the sum of an arithmetic sequence. We know that if we have, if we are taking the sum of, let me do this in a new color, just to have a little bit of variety on the screen, if we're taking the sum of, of the first n terms of an arithmetic sequence or, if we're taking or if we're evaluating the first n terms of an arithmetic series, I could say, it's going to be the first term plus the last term divided by 2. You could view this of the average of the first and the last terms, times the number of terms that we're dealing with. So over here, we know what our first and last terms are, we know this right over here. That is a1, and this is our last term, 2,044, so that is our a-sub-n, so the other question is, well, what is n? How many terms do we actually deal with? And to think about that, we just say, well how many times do we have to add 6 to go from negative 50 to 2,044? Well, 2,044 minus negative 50, minus negative 50, well that's the same thing as 2,044 plus 50 or 2,094, and the whole reason I calculated this is I want to figure out how far do I have to go from negative 50 to 2,044? I have to go up 50 just to get back to zero and then go up another 2,044. So I have to go 50 just to get back to zero, then go up another 2,044 so I have to go up by 2,094, so if I'm going, if I'm adding 6 on every term, how many times do I have to add 6 to increase by 2,094? Well, let's just take 2,094 and divide it by 6 to figure that out. So 6 goes into 20 three times, 3 times 6 is 18, subtract, 20 minus 18 is 2, bring down the 9, 6 goes into 29 four times, 4 times 6 is 24, subtract, 29 minus 24 is 5, bring down the 4, we have 54, 6 goes into 54 nine times, 9 times 6 is 54 and we are done. So to go from negative 50 to 2,044, I have to add 6 to 349 times, so I add it once, I add it twice, and then this right over here, this is the 349th time that I'm adding 6, so how many terms do I have? Now, you might be tempted to say 349 terms, but really, you have 349 plus 1 terms. You have the 349 for every time you added 6, so this is the first time you added 6, second time you added 6, all the way to the 349th time you added 6, so let me make it clear, this, this is, oh, actually, this is the 349th time I added 6 to get to this, but we haven't counted the first term just yet, so we're going to have, so we have 300, we have the first term and then we add 6 349 times, so we have 350 terms in this sum, so in this case, n is going to be equal to 350. N is equal to 350. And so we can say the sum of the first 300... I'll do this in green, the sum of the first 350 terms is going to be equal to the average of the first and last term, so negative 50 plus 2,044 over 2, over 2, times 350, so let's see, negative 50 plus 2,044, that's going to be what? 2,094, 2,094 divided by 2 times, times 350, so let's see, if I just take, so this is going to be, 'cause right, this is 294 times, let's see, 350 divided, oh, sorry, not 294, what am I? My brain is not working. 2,000, this is 2-- actually, this is going to be 1994. My brain really wasn't working a little while ago, so this is going to be 1994 divided by 2 times 350 and so let's see, 350 divided by 2 is 175, so this is going to be 1,900-- 1,994 times 175. Which is equal to, and I'll use a calculator for this one, so, let me get the calculator out, so, I have 1,994 times 175 gives us 348,950. 348,950. And we could express this in sigma notation now, now that we know what the n is. We found our answer, this is what we were looking for, but just in case you're curious, we could write this as the sum from, let's say, k is equal to 1 to k is equal to 350 of, let's see, we could write this as, as negative 50, negative 50 plus 6 times k minus 1, 'cause the first term, we don't want to actually add the 6, and then the last term, we want to add the 6 349 times. Which we saw, so we're going to add it 349 times, and there you have it. That, this arithmetic series written in sigma notation, so hopefully you enjoyed that. Pardon my little mental error earlier. I don't know what was going on in my brain.