Main content

## Algebra 1

### Unit 9: Lesson 3

Introduction to geometric sequences- Intro to geometric sequences
- Extending geometric sequences
- Extend geometric sequences
- Extend geometric sequences: negatives & fractions
- Using explicit formulas of geometric sequences
- Using recursive formulas of geometric sequences
- Use geometric sequence formulas

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Using explicit formulas of geometric sequences

CCSS.Math: ,

Sal finds the 5th term in the geometric sequence whose explicit formula is 3(-¼)^(i-1).

## Want to join the conversation?

- when i raise -1 to the 4th power in my calculator i get -1.Something wrong with my calculator?(3 votes)
- Make sure u have entered it correctly.

If you enter just -1^4, then the calculator will read it as 1 raised to 4th power multiply by -1 i.e. -1.

If you want to do -1 to the 4th power then you should enter (-1)^4. you should get 1.(14 votes)

- This video just wasn't very helpful moving forward because it only gave the answer and steps to answer one specific problem rather than a clear formula for many different problems. Does anyone know the explicit and recursive formulas for geometric sequences, or could someone direct me to another video that I'm not seeing that would be more helpful? Thanks.(2 votes)
- This video explains both explicit and recursive formulas for
*geometric*sequences: https://www.khanacademy.org/math/algebra/sequences/constructing-geometric-sequences/v/explicit-and-recursive-formulas-for-geometric-sequences

These videos explain recursive and explicit formulas for*arithmetic*sequences:

https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/v/recursive-formula-for-arithmetic-sequence

https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/v/explicit-formulas-for-arithmetic-sequences

And these videos explain how to convert between the two:

https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/v/recursive-formulas-for-arithmetic-sequences

https://www.khanacademy.org/math/algebra/sequences/constructing-geometric-sequences/v/converting-an-explicit-function-to-a-recursive-function

Hope this helps!! :)(6 votes)

- is this an example of recursive sequence?(1 vote)
- The video that immediate follows this one covers recursive formulas for a geometric sequence.(2 votes)

- When I solve word problems, they're are not just n-1 for the exponent but also n. When do you put n-1 and n? What do they mean? That's what I don't get. Thank you for the help.(1 vote)
- n stands for the
**N'th**number in a geometric series.

It is very important that you know where the geometric series start, often denoted as**a**.

1) If the series contains a fixed part and a variable part, it is often n-1. E.g. You have 50 euro's in January (month 1) and you add 20 every month. How much do you have in July (7th)? In this case, the**first**addition is in February, which is month 2. Therefore, we need to subtract 1 from the 'the month number'; so it becomes 50+20(n-1) (Note: 30+20n works as well but is not logical to start off with 30).

2) If the first term is part of a larger series; like 3,9,27,81,243,729. The formula 3^n would make sense. Since the first number is n=1 (in math we often start at 1; in coding usually at 0), 3^1=3 is correct.(1 vote)

- I dontunderstand how to figure them out(1 vote)
- why didnt he multiply the 3 to 256(1 vote)
- When I put this into my calculator, it showed the answer, but it was negative. I need to know if I have done something wrong.(1 vote)
- You likely typed in: -1^4 when you needed (-1)^4.

You need to parentheses to indicate that the base of the exponents is the negative number. If you don't use the parentheses, then the base is a positive value and the "-" is applied after the exponents.

Hope this helps.(1 vote)

- I got a problem that went like

c(n)=4/9(-3)^n-1

trying to figure out the third term in the sequence

4/9(-3)^3-1 to 4/9(-3)^2 = 4/9*6 =2.66...

but the hint said that the answer was 4 and that would need the exponent to be 3. Why didn't the subtract the exponents?(1 vote)- The error is in your multiplication.

(-3)^2 = (-3) times (-3) = +9, not 6

Hope this helps.(1 vote)

- how did you get the answer(1 vote)
- the exponent wasn't multiplied from outside the bracket ?(1 vote)
- The 5-1 =4 was multiplied from outside the bracket, and the way to do this is to distribute the 4 to all the powers inside. Since 1/4 is to the first power (1/4)^1, when you multiply 1 * 4, you get 4 inside the bracket. Hope this answers your question.(1 vote)

## Video transcript

- [Voiceover] The geometric
sequence A sub I is defined by the formula and so they
tell us that the Ith term is going to be equal to three
times negative one fourth to the I minus one power. So given that, what is A sub five, the fifth term in the sequence? So pause the video and try to figure out what is A subscript five? Alright, well, we can
just use this formula. A... A sub five is going to be... is everywhere I see an
I or a place with a five is going to be equal to three
times negative one fourth to the five minus one power. Well that's equal to three
times negative one fourth to the fourth power. Well that's going to be equal to... lets see, we're raising
it to an even power so it's going to give us a positive value since we're gonna be
multiplying the negative an even number of times so
it's gonna be a positive value so it's gonna be three times... let's see, one to the one fourth is- oh, one to the fourth power is just one, and then four to the fourth power... let's see, four squared is 16, so four squared times four
squared is four to the fourth so it's 16 times 16 is 256. 256. And once again I know
it's going to be positive because I'm multiplying a
negative times itself four times, or I'm multiplying four
negatives together, so that's going to give
me a positive value. So I get three over 256. And we're done. That's the fifth term in our sequence. Positive three over 256.