Question

What is the 10th term of the sequence 81, 27, 9,...

Answer 1

A. 1/179
B. 1/243
C. 1/81
D. 1/810

Answer 2

What's your answer?

Answer 3

I think it's D

Answer 4

Nope it isn't D

Answer 5

Hmm...let me work it again I might have gone to far

Answer 6

C. 1/81?

Answer 7

Nope

Answer 8

Wait that's 9th term it's 1/243

Answer 9

I strongly recommend that you find the appropriate expression for the nth term of this geometric sequence.

Answer 10

Yep that's correct

Answer 11

What is the initial value, a?
What is the common ratio, r?

Answer 12

By the way do you understand the concept?

Answer 13

Yea it was division by 3

Answer 14

Or, "our common ratio is 1/3."

Answer 15

so, r = 1/3.

What is the first term, a? a = ?

Answer 16

a = 81 (from the sequence it was given)

Answer 17

Right. Could you now develop a formula for the nth term of this geometric sequence?

\[a _{n+1}= ?\]

Answer 18

a(1/3) + 1 = ? (would I just put 81 then 27 then 9 etc.. in the a term?)

Answer 19

How would you find the next term of this sequence? Describe the process by words please.

Answer 20

What we need here is a formula in n (that means that n is your independent variable) that correctly predicts the nth term of this geometric sequence. Your tentative result does not involve n, so is not correct.

Look: If n=0, a*(1/3)^0=81*(1/3)^0 = ?

Answer 21

81*(1/3)^0 would equal 81?

Answer 22

ok ... 81/3 = 27 27/3 = 9
Do that a few more times.................. its going to get negative.

Answer 23

There are two presentations for a formula of a geometric sequence

Answer 24

or... Arithmetic sequence:1, 3, 5, 7
Notice how I add 2 everytime. This is arithmetic.1, 2, 4, 8
This is geometric. I am not adding or subtracting the same number every time, as 1 to 2 is +1, while 2 to 4 is +2. So what is the pattern? It is geometric. I am multiplying by 2 to get my next sequence.

So in your problem, you are not adding or subtracting the same number every time because clearly this is not the case. So this is NOT an arithmetic sequence, but if you can find a common multiplying or dividing factor you can identify it as geometric. As others have mentioned, you are dividing by 3 (or multiplying by 1/3) each time to obtain your next time.

Answer 25

One is called recursive formula, which is recursive (thank you captain obvious)

Answer 26

One is called explicit formula or closed-form expression, which is not recursive.

Answer 27

So this would be an explicit formula?

Answer 28

What do you mean by this?

Answer 29

Nvm XD Thought you where wanting me to choose which formula to use

Answer 30

An explicit formula for a geometric sequence would be A_n = a * r^(n-1)

Answer 31

Or in LaTeX, \(\Large A_n=a\cdot r^{n-1}\).

Answer 32

Where \(A_n\) is the n-th term, a is the first term, and r is the common ratio

Answer 33

81*(1/3)^0 would equal 81? YES.
What about 81*(1/3)^1?

What about 81*(1/3)^2?

You'll see, through experimentation, that this fits Kenny's explicit formula perfectly.

Answer 34

A general recursive formula for a geometric sequence would be A_(n+1) = r * A_n

Answer 35

Or in LaTeX, \(\Large A_{n+1}=r\cdot A_n\).

Answer 36

If you agree, take Kenny's formula and substitute n=10. What's the 10th term?

Answer 37

Where \(A_{n+1}\) is the next term, \(A_n\) is the previous term, and \(r\) is the common ratio.

Answer 38

Also, @Cpt.Sparz, would you look at Kenny's two different formulas and decide which one would be better suited for use in predicting the 10th term of this geom. seq.?

Answer 39

And why?

Answer 40

(explicit formula) 81*1/3^10 - 1 = 1/243

Answer 41

Explicit formula comes easier to me

Answer 42

That'd be my choice also. Note that your response would be better if written as\[81*(1/3)^{10-1} = 1/243\]

Answer 43

Oh ok...I don't know how to work that :3

Answer 44

Like how to make it look that...

Answer 45

I'd suggest learning how to use Equation Editor, if you haven't already.
I'd choose the explicit formula because we need only substitute n=10 there, whereas in the case of the recursion formula, you have to calculate all 10 terms of the sequence.

Answer 46

You only have to add in parentheses.

Answer 47

81*1/3^(10 - 1) = 1/243 would be so much better than 81*1/3^10 - 1 = 1/243

Answer 48

Don't have to use the equation editor so often :p

Answer 49

Oh ok thanks @kc_kennylau @mathmale

Answer 50

\[81 * (\frac{ 1 }{ 3 })^{10 - 1} = \frac{ 1 }{ 243 }\]

Answer 51

That's better

Answer 52

Note that \[81*(1/3)^{10-1} = 1/243\] shows that you're on the right track, but 1/243 is not the correct 10th term.

\[81*(1/3)^{10-1} = 81*(\frac{ 1 }{ 3 })^{9}=\frac{ 3^4 }{ 3^9 }=??\]

Answer 53

Note that 81=3^4.

Answer 54

How would you reduce \[\frac{ 3^4 }{ 3^9 }?\]

Answer 55

\[\frac{ 3^{4} }{ 3^{9} } = \frac{ 1 }{ 243 }\]

Answer 56

Right. Sorry, I was wrong; 1/243 was correct all along. But this gives you an alternative way of evaluating the 10th term.

Answer 57

Yea! I've never seen this way before, but I like it...more neat!

Answer 58

cool. Hope to work with you again sometime soon! Bye for now.

Answer 59

Thanks again!