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### Course: Algebra 2 (Eureka Math/EngageNY)>Unit 3

Lesson 12: Topic E: Lesson 29: Geometric series

# Finite geometric series formula

A finite geometric series can be solved using the formula a(1-rⁿ)/(1-r). Sal demonstrates how to derive a formula for the sum of the first 'n' terms of such a series, emphasizing the importance of understanding the number of terms being summed.

## Want to join the conversation?

• At , Sal multiplies ar^(n-1) by -r and gets -ar^n. I do not quite get how that works and would like some help on it.

• OK, this is a really REALLY great question. When you multiply ar^(n-1) and -r together the first thing you can do is distribute the negative sign, which gives you -ar^(n-1) * r. The variable r can also be expressed as r^1. So you get -ar^(n-1) * r^1. Next you can pull out the -a which gives you (-a)(r^(n-1)) * r^1. Then you can simplify and get (-a)(r^(n-1+1)). Once again that can be simplified very easily to
-ar^n. I hope that was helpful.
• Great, so, that's the formula. Simple. But WHY? Why does this formula give us the sum? Does anyone know of any videos anywhere that actually explain WHY this works? And where it came from? Sal said "We're going to think about what r times the sum is and then subtract that out" but never gave an explanation.

ETA: If anyone's interested, I just found an awesome vid on Eddie Woo's youtube channel that goes into more detail on why this works, it's called: Intro to Geometric Progressions (2 of 3: Algebraic derivation of sum formula)
• This is just what I came here to post. It's as if we're supposed to just say, "cool thanks for the formula Sal!" and walk away without actually understanding what exactly is going on under the hood here... Thank you for the link I'm going to check it out now
• Why is it that I was watching a video in which he says he already derived this formula, and when I finally found the video where he derives it, it's located after the video I was watching, in which it was assumed I knew this formula.
• This lessson should be placed higher up right after "Geometric series with sigma notation" because in the video lesson following "Worked example:finite geometric series(sigma notation) it says the general formula was already mentioned in a previous video when it was not.It is only mentioned in this last video lesson "Finite geometric series with formla justification".
Please correct this mistake as it is confusing.
• this is where I still struggle... how do I know to multiply by -r and then add the resulting equation to the original? I guess it's just, well that, a guess and it's "intuition", but then... my question is how do I get to build that intuition so that I can do it myself for other things?
• Practice helps build intuition, now for an endless amount of series to practice with I can only highly recommend pascal's triangle, and using its "diagonals" as series and trying to figure out the formula for each of them.

Here's a picture of pascal's triangle, and the "diagonals" are highlighted http://www.mathsisfun.com/images/pascals-triangle-2.gif
For each consecutive row you add the number on the left and the right on the rows above to get your number, and a blank = 0... I can't explain it properly but its super easy, so here how it goes :
Row 1 = 1
row 2 = 0+1 , 1+0 = 1 , 1
row 3 = 0+1 , 1+1 , 1+0 = 1 , 2 , 1
row 4 = 0+1, 1+2, 2+1, 1+0 = 1 , 3 , 3 , 1
etc

The diagonals are:
D1 = 1, 1, 1, 1, 1, 1, 1, 1 ...
D2= 1, 2, 3, 4, 5, 6...
D3= 1, 3, 6, 10, 15 ...
etc
Try taking the sum of these series, and make a function for each of them, and then find a generic formula for all the diagonals if you're feeling brave!

A tip i can give you, is to try to go from something you don't know to something you do know, the path between the two is "intuition".

And as a bonus, pascal's triangle has way more than just series, try exploring it and figuring out its properties, it's fascinating ! By doing so, you'll be building up your "intuition", I can guarantee it! if the greeks had known about it, they'd have built temples and revered it like a deity.
• Is it possible to find n by using a formula, as it is with arithmetic series?
• The video is actually about geometric series, however it is useful some knowledge regarding arithmetic series.

It will depend on the exact question.

How many number are there from 0-150?

Ans: 150 - 0 + 1 = 151

There is the plus one because we need to include 0.

How many numbers are there in the given sequence:

0, 2, 4, ...., 20

If we divide by 2 we get:

0, 1, 2, ..., 10:

Ans: 10 - 0 + 1 = 11 numbers

How many numbers are there in the sequence:

7, 9, 11, ..., 21

Subtract by 7 to get:

0, 2, 4,..., 14

Divide by 2:

0, 1, 2, ..., 7

Therefore the amount of numbers is 7-0+1 = 8
• Is there a name for this technique of finding a formula?
• Converting between a recursive form and an explicit form of an expression?
• QUESTION

ok... I'm so confused! s of n? a's and r's? I have no idea what's going on, help? I would appreciate it! thanks! :D
• So the majority of that video is the explanation of how the formula is derived. But this is the formula, explained:

Sₙ = a(1-rⁿ)/1-r

Sₙ = The sum of the geometric series. (If the n confuses you, it's simply for notation. You don't have to plug anything in, it's just to show and provide emphasis of the series.

a = First term of the series
r = the common ratio
n (exponent) = number of terms.

As an example:
What is the sum of the 4,16,64,256?
The common ratio is 4, as 4 x 4 is 16, 16*4 = 64, and so on.
The first term is 4, as it is the first term that is expliicty said.
There are 4 terms overall.

Plugging it into the formula...

Sₙ = 4(1-4⁴)/1-4 = 4(-255)/-3 = -1020/-3 = 340

Why do we use this ? This is just an easy example, some series can be absolutely crazy – this is what the series are for.

Hopefully that helps ! I only specified what the formula is and how it's used, not the background of it.
• How am I supposed to memorize this formula? I understand everything in the video but it just doesn't stick to my head. I was reviewing this lesson a few weeks after I learned it and I didn't remember anything. Also, when am I supposed to use this formula? I don't really understand its purpose.
• You don't need to memorize it. Practice questions involving this formula and you'll eventually remember it. Plus, the derivation isn't too hard, so even if you forgot the formula, you can just derive it.

The formula is used a lot in infinite series, where we have infinite geometric series which converge. There, you use a slightly modified version of this formula to find the sum. This formula itself is used for, as the video says, finding the sum of a finite geometric series