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## Algebra 2 (Eureka Math/EngageNY)

### Unit 3: Lesson 13

Topic E: Lesson 29: Arithmetic series- Arithmetic series intro
- Arithmetic series formula
- Arithmetic series
- Worked example: arithmetic series (sigma notation)
- Worked example: arithmetic series (sum expression)
- Worked example: arithmetic series (recursive formula)
- Arithmetic series worksheet
- Arithmetic series
- Proof of finite arithmetic series formula

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# Arithmetic series

Walk through a guided practice where you'll start by finding a simple sum and end by evaluating finite arithmetic series.

Let's start with an addition problem.

### Find the sum of 1, plus, 3, plus, 5, plus, 7, plus, 9.

Awesome! You just found the sum of a small arithmetic series. It only had 5 terms. But, what if it had one million terms? We'd want a formula for sure. Thankfully, we've already learned of such a formula.

**Identify the formula for the sum of an arithmetic series.**

Sweet! So you remember the formula. Now let's make sure we remember how to apply it.

**Choose the answer that shows the formula correctly used to find the sum you found.**

Alright, so we're feeling good so far. Let's try to use the formula to find the sum of an arithmetic series that would be tedious to calculate by hand.

## Consider the series 3, plus, 5, plus, 7, plus, point, point, point, plus, 401.

**Find the values of a, start subscript, 1, end subscript and a, start subscript, n, end subscript for this series.**

**Find the value of n for this series.**

**Find the sum of 3, plus, 5, plus, 7, plus, point, point, point, plus, 401**

Wow! Okay, looks like you've got this.

## Try it yourself

Nice! Try another one!

## Want to join the conversation?

- I do not understand how to find the n value. This did not make any sense to me. I need a formula or an explanation on how to find the n.(34 votes)
- Another way you can do it is to come up with a function, and plug in your final value and solve for x. For example, when finding the sum of 3 + 5 + 7 + .... + 401 It might help to start out with a little chart:

x f(x)

1 3

2 5

3 7

4 9

And from that you might intuit

f(x) = 2x + 1

Then plug in your final term, which is 401, and solve for x

401 = 2x + 1

400 = 2x

x = 200 tada

Sometimes you’ll see a slightly different, but equivalent function. For example, the same 3 + 5 + 7 + .... + 401 you might first identify as

f(x) = 3 + 2(x-1)

and that’s fine, because mathematically it’s the same function. Even if you don’t simplify it to 2x + 1 you still find x = 200 when you plug in your final term in the sequence.

401 = 3 + 2(x-1)

401-3 = 2(x-1)

398 = 2 (x-1)

398/2 = x-1

199 = x - 1

x = 200(7 votes)

- How do we calculate the value of n(5 votes)
- Take the LAST number in the sequence MINUS the FIRST number in the sequence.

DIVIDE that value by the pattern in the sequence.

How do you find the pattern? Ask, "How do I get from the first term, to the second term?"

Hope that helps(22 votes)

- How do you determine the value of n? I don't think this was every explained.(9 votes)
- n is the number of terms. Take the difference of the first and last term over the common difference and add 1 to get n.(8 votes)

- What's exactly the difference between "progression", "sequence" and "series"?(6 votes)
- Progession and sequence are the same thing; a list of numbers generated according to some rule or rules.

For example 2,4,6,8,10 is an (arithmetic) sequence.

Or 1, 2, 4, 8, 16, which is a geometric sequence.

A series however is the SUM of a sequence or progression

eg 1 + ½ + ¼ + ⅛

There are a number of KA videos on this subject, here's one: https://www.khanacademy.org/math/algebra2/sequences-and-series/copy-of-geometric-sequence-series/v/series-as-sum-of-sequence(8 votes)

- For problem 2: how did you find out the n was 450? I tried doing An-A1/2 and adding one for the first term and I got a different number.(7 votes)
- it's like this...

let's say we have an arithmetic sequence that goes like 2, 4, 6, 8, ..., 262

lets take first term as a=2

common difference as d=2

last term or nth term as a(n)=262

we know the formula for nth term is a(n)=a+(n-1)d

so here, 262=2+(n-1)2

so n-1 = (262-2)/2 = 260/2 = 130

so n= 130+1 =131

Sal explained it in a non-formula, less mathematical more logic-based kind of way, but this is the mathematical basis for it

hope it helps :)(1 vote)

- How is it 1000 for the last one?(6 votes)
- i got 997.09, i think kahn academy made a mistake(0 votes)

- can somebody pleeeease explain how to find n like in question 4b I read the comments and the explanation but couldn't conclude a formula(4 votes)
- it's like this...

let's say we have an arithmetic sequence that goes like 2, 4, 6, 8, ..., 262

lets take first term as a=2

common difference as d=2

last term or nth term as a(n)=262

we know the formula for nth term is a(n)=a+(n-1)d

so here, 262=2+(n-1)2

so n-1 = (262-2)/2 = 260/2 = 130

so n= 130+1 =131

Sal explained it in a non-formula, less mathematical more logic-based kind of way, but this is the mathematical basis for it

hope it helps :)(1 vote)

- on problem 1, could someone tell me how they found out the number of terms was 450?(2 votes)
- 11+20+29+...+4052

They found n (the last term) by set 4052 into the explicit formula. So to find n you must know how to formulate the formula from the sequence.

This has the initial of 11 and common difference of 9, so a(n)=11+9(n-1). So to find what n is when a(n) = 4052, you set 4052=11+9(n-1) and solve for n. They didn't explain but that's how you would find n in that problem.(4 votes)

- What the heck does this mean: Find the sum of first 335 terms

A(sub)1 = 2

A(sub)i = A(sub< i -1 >) -3

What does the "i" mean?(2 votes)- The subscripted numbers denote the number of the term in series A. The subscripted i can be any number other than 1. (The first term is separately defined.) The subscripted i-1 refers to the term immediately before term i.

Based on the example you gave, here are the first few terms. (The underscore is used to show a subscript.)

A_1 = 2

A_2 = A_1 - 3 = 2 - 3 = -1

A_3 = A_2 - 3 = -1 - 3 = -4

A_4 = A_3 - 3 = -4 - 3 = -7(4 votes)

- I did not understand this sum:

Find the sum; 11+20+29+...+4052.

Please let me know,how to solve this?(2 votes)- In order to solve this series we have to make pairs.

1. Find the number of total numbers in the equation.

So because we start at 11 we subtract 11 from 4052. Then we get 4041, Next we divide by 9 because that is the increments that this is increasing by. This gives us 449 but we must remember that at the beginning we took out the 11, so we have 450 total numbers. (Wow what are the chances of getting such a nice number😮)

2. Next we add the smallest and biggest numbers together. We do this 225 times. The smallest and biggest numbers are 11 and 4041. when we add them together we get 4063. We get this number 225 times because after 11 and 4052 are used we go to the next numbers 20 and 4043. You will see that we will get the same number every time because the small number increases by 9 and the big number decreases by 9.

3. Now we just need to do some simple arithmetic. 4063*225= 914175

4. Check the [Explain] button before you ask a question because it says what I am saying here but with equations instead of words.(4 votes)