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## Algebra (all content)

### Course: Algebra (all content)>Unit 18

Lesson 2: Arithmetic series

# Arithmetic series intro

Sal explains the formula for the sum of a finite arithmetic series. Created by Sal Khan.

## Want to join the conversation?

• How do you figure out the sum of an alternating sequence like the sum of the first 39 terms of
1 + (-1)^n?
• You can look at it as a sum of two sequences--the first is arithmetic, with initial term a=1 and term difference of d=0. The first term is 1, the 39th ("last") term is 1+0*39=1. Sum of an arithmetic sequence is (first + last)*(#terms/2) = (1+1)*(39/2) = (2)*(39/2) = 39. The second sequence is geometric, with initial term a=-1 and term ratio r=-1. Sum of a geometric series, from another video, is a*(1-r^n)/(1-r)

= (-1)*(1-(-1)^39)/(1-(-1))
= (-1)*(1-(-1))/(1-(-1))
= (-1)*(1+1)/(1+1)
= (-1)*2/2
= -1

You can verify this intuitively by considering that the first term is -1, and then the next 38 terms cancel each other out in pairs (2nd and 3rd cancel, 4th and 5th cancel...38th and 39th cancel).

Sum the two results to get 39 + (-1) = 38.

Not every sequence can be so simply broken down into a sum of other (simple) sequences, but this one can.
• At how is 2+(n-1)=(n+1)? That doesn't make any sense to me.
• 2 + 𝑛 – 1
We can combine the 2 and the -1 to get 2 – 1 = 1. Thus:
2 + 𝑛 – 1 = 𝑛 + 1
Proficiency in this type of algebra is expected for studying arithmetic series. I would recommend looking at the "solving one step equations" playlist and the "combining like terms" video.
• If this is true, why is the sum of all positive integers -1/12?
• It's important to keep in mind that the people who make that series add up to -1/12 are willfully breaking some of the rules of standard mathematics (and usually not mentioning that fact). For the purposes of what they're doing (unless they're just doing it to freak out the newbies), they're not wrong, but it just makes things confusing to people learning the rules of standard math. For what you're learning now, the intuition that the sum of all positive integers should be undefinable because it is infinitely large is the correct attitude.
• how do you write the sigma notation for the sequence 1+5+9+13+17+21 what is n? and what would the general form be?
• nth term: 1 + 4 * ( n - 1 ) = 1 + 4n - 4 = 4n - 3
6
Σ ( 4n - 3 ) = 1 + 5 + 9 + 13 + 17 + 21
n = 1
• So how would you find the sum of an infinite geometric series?
• You cannot add up an infinite number of numbers, but you can take the limit of the sum as n approaches infinity. Sometimes this gives you a number, sometimes it gives you infinity, sometimes it isn't helpful at all. Much of the time, you must be content to know if the sum "converges" or "diverges".
You'll learn about this in calculus. Unfortunately, Sal only has a couple of videos on the basics of this broad subject. Try a Google search for "Infinite Series".
• What does the n stand for?
• "n" is used to represent the term in the sequence. If you need the 3rd term, then n=3. If you need the 10th term, the n=10.
Hope this helps.
• Isn't that called the "little Gauss" after the German mathematician Carl Friedrich Gauss, who discovered that formula in elementary school? Maybe the story is apocryphal...
• at , so the n has 2 meanings. One means a (sub n), and the other means the number of terms?
• "n" always represents the number of the term. "a(sub n)" refers to the value of sequence at term "n". For example, if n=3, then...
a(sub 3) = the value of the sequence that is the 3rd term
n = 3 says you are finding / using the 3rd term.
• Hi I just wanted to ask what does n-1 and n-2 mean
• Hi! Consider series 1,2,3,4,5 for example.
There are 5 terms in this Arithmetic series.
The fifth term is 5 and it is n
The fourth term is 4 and it is n-1
The third term is 3 and it is n-2
The second term is 2 and it is n-3
The first term is 1 and it is n-4. We know that n = 5 so n-4 is 1, n-3=2....