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### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 3: Series review

# Worked example: arithmetic series (sigma notation)

Sal evaluates the arithmetic series Σ(2k+50) for k=1 to 550. He does that using the arithmetic series formula (a₁+aₙ)*n/2.

## Want to join the conversation?

• Is that formula the same for all sigma notation problems?
• I'm assuming you are referring to the formula for the sum of a finite arithmetic series, which Sal defines starting at around . If that's the formula you mean, then: No, it isn't the same for all finite sums: The formula Sal uses will work only for arithmetic series. Sigma notation is used for all kinds of sums, and not just arithmetic series. Different series will have different sum formulas.
• I checked that it is an arithmetic sequence first by seeing that d( 2*x + 50 ) / dx = 2 which is a constant. Thus the sequence terms are changing by a constant. Does this mean that we get an arithmetic series whenever the expression is linear with respect to the index ?
• Yes, because the "𝑥:th" term of an arithmetic sum is always 𝑡(𝑥) = 𝑎 + 𝑏𝑥, where 𝑎 = 𝑡(1) and 𝑏 is the difference between two consecutive terms, 𝑏 = 𝑡(𝑘 + 1) – 𝑡(𝑘).

This means that the sigma notation will be 𝛴(𝑎 + 𝑏𝑥), 𝑥 = 0 → 𝑛 – 1, where 𝑛 is the total number of terms.

Even if we changed the notation to, for example, 𝛴(𝑎 + 𝑏(𝑥 – 1)), 𝑥 = 1 → 𝑛 ⇒
⇒ 𝑑/𝑑𝑥 ∙ (𝑎 + 𝑏(𝑥 – 1)) = 𝑏, so it still holds up.

And, since these are all definitions of an arithmetic sum, there can't be any other types of sums that match this relationship.
• Why does Sal add two sums of S(n)?
• I believe he was trying to show us why we have (n/2) and how the final answer comes about. He explains where the /2 comes from in the Arithmetic Series Formula video so if the formula confuses you best to probably rewatch that video, but as you can he gives you the answer at so everything else after that is just him proving it to us.
• Well, I what I don't get is that in the video 'Worked example: arithmetic series (recursive formula)' the rule is that it increases with a + 11. And a = 4. But in that video Sal said the 1st term would be 4, not 15 which is 4+11. How come here he actually works out the 1st term using the rule so the 1st term is 52 which K = 1 *2 + 50? Why can't he just be consistent? :(
• It's not a matter of being consistent. He's right. The first term in Worked example: arithmetic series (recursive formula) is 4. Then the second term was 15. This is because it's a recursive. He was given that the first term is 4. Now for this video, it's the sum of 2k + 50 from index k =1 to k=550. Since the index starts at k =1, he has to plug in 1 to find the first term so, it's 52. If you were thinking that you plug in 0 to find the first term is 50, you can't do that because the index starts at k =1. Hope this helps. If you need further clarification let me know.
• Does n in the formula for S_n strictly represent the number of terms being summed, or the value of the upper index? I'm asking because I've noticed that the starting index doesn't have to be 1.
• In the event of S(n), "n" is going to refer to the number in a series, and nth term of it. In the case of sums of series, you will commonly see sigma notation used, where the starting term does not have to be the first term in a series. In that case, expect to find the the number of the term that "starts" the addition below the sigma, and the final term on top.
• In this example, "k" is 1-550, which means "n" of the sum formula would be 550 terms, correct? If "k" was 0-550, would "n" be 551 terms?
• k varies, starting at 1 and going up to 550. How many values of k is that?

If j varies from 1 to 5, how many values does it take?

(5.)

So therefore, in the video, k takes 550 values.

If j goes from 0 to 5, how many values does it take on?

(0, 1, 2, 3, 4, 5; that is, 6 values.)

So if k goes from 0 to 550, yes, that's 551 values.

B u t, don't forget that n is still 550, not 551.
• Isn't this formula that the legendary mathematician Gauss used in elementary school to shock his teacher?
• Yeah, the story goes that the teacher told the students to calculate 1 + 2 + 3 + ... + 100, which the teacher thought would take a while, but Gauss (10 years old at the time) quickly realized that it added up to 101 ∙ 50 = 5050, which, needless to say, startled the teacher.
• Hey guys, I need some help! How do I simplify this?
An auditorium can fit 252 people. If the first row fits 16 people and the following rows can fit 3 more people than the preceding row, how many rows are there in the auditorium?
Sn = n(a1+an)/2
252 = n(16 + an)/2
Because I don't know a sub n, I can't find n, so I have to plug in numbers till I find the answer:
16+19+22+25+28+31+34+37+40=252
Thus, there are 9 rows. Is there an equation I can use to solve for this without plugging in numbers?
• One method is to get an in terms of n. Since the first term is 16, and each term after the first is 3 more than the previous term, it follows that the nth term is an = 16 + 3(n - 1).

So we have the quadratic equation 252 = n[16 + 16 + 3(n - 1)]/2, which we can then solve for n. While the quadratic could be solved by factoring, using the quadratic formula is probably easier in this situation. Note that we would disregard any solution that is negative or fractional, because the number of rows, n, must be a non-negative integer.

Have a blessed, wonderful day!