does this finite geo series formula always work? or is there a specific restriction for the variables?

Geometric series always work except where common ratio = r = 1, since the denominator will be zero. But when r = 1, you can easily claim that all terms in the series are equal. Other than that, the geometric series will always work.

Can anyone explain geometric series in simple lang

A geometric series is a sequence of numbers that follow a pattern where each term is obtained by multiplying the previous term by a fixed number. For example, the sequence 1, 2, 4, 8, 16, … is a geometric series because each term is twice the previous term. The fixed number that we multiply by is called the common ratio. The formula for finding the sum of an infinite geometric series is a / (1 - r), where a is the first term and r is the common ratio. If |r| < 1, then the sum of the series is finite and can be calculated using this formula. If |r| >= 1, then the series diverges and does not have a finite sum. For example, consider the geometric series 1 + 2 + 4 + 8 + .... Here, a = 1 and r = 2. Since |r| >= 1, this series diverges and does not have a finite sum.

Why is it in Geometric sequences is the 0 term just the first number? Ex: 2,2+3,2+3^2,.... Why isnt it 2+1,2+3,2+3^2? This makes sense because anything to the 0 power is 1, so if we are making the exponent 0, wouldnt it just be 1?

The general form of the nth term of geometric sequence is ar^(n-1). Now, the first term would indicate n = 1. So, we get ar^(1-1) = ar^(0) = a. That's why the first term is just the first number.

In deriving the geo series formula, why are the 2 lines added together? What is the reason? Tnx!

They are equivalent expressions, and since they are equivalent, when you add one to the other, you are adding the same thing to both sides. e.g. a = 12 now lets square both sides to make another equation a^2 = 12^2 now add both equations a + a^2 = 12 + 12^2 it's still equivalent. Seems pointless here but as you saw from the video and you will see in the future, this allows us to manipulate equations in ways we couldn't do otherwise (e.g. in the geometric series, it allows us to add/subtract an infinitely repeating series to get a formula to solve for the variables in the series).

How can you factorise n^4+6n^3+11n^2+6n+1?

You can factor this equation out as (n^2+3n+1)^2

Main content

Algebra 2

Course: Algebra 2 > Unit 3

Lesson 7: Geometric series

Polynomial factorization: FAQ

Google Classroom

Frequently asked questions about polynomial factorization

What is factoring?

Factoring is the process of breaking down a polynomial into smaller pieces (or "factors") that, when multiplied together, will give you the original polynomial.

Why do we factor polynomials?

Factoring is a useful technique for solving polynomial equations. By breaking a polynomial down into smaller factors, we can often simplify the equation and find the solutions more easily.

What is a monomial?

A monomial is a polynomial with just one term. For example,

2 x

- 3 y^{2}

, and

5

are all monomials.

What is the greatest common factor?

The greatest common factor (GCF) of two or more terms is the largest factor that all the terms have in common. For example, the GCF of

6 x^{2}

and

9 x

3 x

How do we take common factors out of a polynomial?

To take common factors out of a polynomial, we divide each term by the GCF. For example, to factor

6 x^{2} + 9 x

, we divide both terms by the GCF,

3 x

, to get

3 x (2 x + 3)

What are polynomial identities?

Polynomial identities are equations that are true for all values of the variable. For example, the difference of squares identity,

a^{2} - b^{2} = (a + b) (a - b)

, is true for any values of

a

and

b

Where are these topics used in the real world?

There are many applications for factoring polynomials. For example, engineers may use these techniques in signal processing or control theory, while scientists might use them in modeling physical phenomena.

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26huangd
Posted 5 months ago. Direct link to 26huangd's post “does this finite geo seri...”
does this finite geo series formula always work? or is there a specific restriction for the variables?
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(2 votes)
Answer
- joshua
  Posted 5 months ago. Direct link to joshua's post “Geometric series always w...”
  Geometric series always work except where common ratio = r = 1, since the denominator will be zero. But when r = 1, you can easily claim that all terms in the series are equal.
  
  Other than that, the geometric series will always work.
  Button navigates to signup page
  (8 votes)
zaaraz.fashion
Posted 7 months ago. Direct link to zaaraz.fashion's post “Can anyone explain geomet...”
Can anyone explain geometric series in simple lang
Button navigates to signup pageComment on zaaraz.fashion's post “Can anyone explain geomet...”
(2 votes)
Answer
- bingai
  Posted 2 months ago. Direct link to bingai's post “A geometric series is a s...”
  A geometric series is a sequence of numbers that follow a pattern where each term is obtained by multiplying the previous term by a fixed number. For example, the sequence 1, 2, 4, 8, 16, … is a geometric series because each term is twice the previous term. The fixed number that we multiply by is called the common ratio.
  
  The formula for finding the sum of an infinite geometric series is a / (1 - r), where a is the first term and r is the common ratio. If |r| < 1, then the sum of the series is finite and can be calculated using this formula. If |r| >= 1, then the series diverges and does not have a finite sum.
  
  For example, consider the geometric series 1 + 2 + 4 + 8 + .... Here, a = 1 and r = 2. Since |r| >= 1, this series diverges and does not have a finite sum.
  Button navigates to signup page
  (3 votes)
Noma Woodall
Posted 5 months ago. Direct link to Noma Woodall's post “how do you find the minim...”
how do you find the minimum of a degree on a graph?
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(3 votes)
Answer
- dashpointdash
  Posted 5 months ago. Direct link to dashpointdash's post “I guess one has to basica...”
  I guess one has to basically reconstruct the function or try to fit another function that is similar or use a Taylor series expansion, which I guess is mentioned in college algebra.
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  (0 votes)
Nicole
Posted a month ago. Direct link to Nicole's post “why is it when r is a per...”
why is it when r is a percentage, it can be written as 1.the percentage or 0.the percentage? for example 10% was converted to 1.1 and not 0.1 but 90% was 0.9 and not 1.9
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eshanpradhan0115
Posted 4 months ago. Direct link to eshanpradhan0115's post “for the earlier section w...”
for the earlier section we used the formula where "a" was included numerator section but now im using it like this?
a(1-r^n/1-r)?
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(2 votes)
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- David Li
  Posted 4 months ago. Direct link to David Li's post “Both is correct, just two...”
  Both is correct, just two different ways of representation
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  (1 vote)
McKinnon Bilal
Posted 7 months ago. Direct link to McKinnon Bilal's post “Why is it in Geometric se...”
Why is it in Geometric sequences is the 0 term just the first number?

Ex: 2,2+3,2+3^2,....

Why isnt it 2+1,2+3,2+3^2?

This makes sense because anything to the 0 power is 1, so if we are making the exponent 0, wouldnt it just be 1?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Venkata
  Posted 7 months ago. Direct link to Venkata's post “The general form of the n...”
  The general form of the nth term of geometric sequence is ar^(n-1). Now, the first term would indicate n = 1. So, we get ar^(1-1) = ar^(0) = a. That's why the first term is just the first number.
  Comment on Venkata's post “The general form of the n...”
  (2 votes)
Greg One
Posted 6 months ago. Direct link to Greg One's post “In deriving the geo serie...”
In deriving the geo series formula, why are the 2 lines added together? What is the reason? Tnx!
Button navigates to signup pageComment on Greg One's post “In deriving the geo serie...”
(1 vote)
Answer
- Lotup
  Posted 5 months ago. Direct link to Lotup's post “They are equivalent expre...”
  They are equivalent expressions, and since they are equivalent, when you add one to the other, you are adding the same thing to both sides.
  
  e.g.
  
  a = 12
  now lets square both sides to make another equation
  a^2 = 12^2
  now add both equations
  a + a^2 = 12 + 12^2
  
  it's still equivalent. Seems pointless here but as you saw from the video and you will see in the future, this allows us to manipulate equations in ways we couldn't do otherwise (e.g. in the geometric series, it allows us to add/subtract an infinitely repeating series to get a formula to solve for the variables in the series).
  Button navigates to signup page
  (2 votes)
shan.n.shanmu
Posted a year ago. Direct link to shan.n.shanmu's post “How can you factorise n^4...”
How can you factorise n^4+6n^3+11n^2+6n+1?
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(0 votes)
Answer
- kazashr000
  Posted a year ago. Direct link to kazashr000's post “You can factor this equat...”
  You can factor this equation out as (n^2+3n+1)^2
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  (3 votes)