Question

A recursive rule for a geometric sequence is a1=4/9; an=3(an-1)

What is an explicit rule for this sequence?

Answer 1

An =?

Answer 2

Are you familiar with that?

Answer 3

Yes

Answer 4

But when it comes to recursive terms, its a little confusing for me

Answer 5

No worries..
Step by step.

\[\Huge a_1=\frac{ 4 }{ 9 }~~~,~~~ a_n=3*a_{n-1}\]

That means two things:
- the first term in this sequence is 4/9
- the common ration is 3, i.e to get any term you have to multiply the previous term by 3 to get the new term


agree so far?

Answer 6

Yes, I catch on so far

Answer 7

What do I do next?

Answer 8

That is great!

Now for the explicit rule for this sequence:
you need to type out the first term \(\color{red}{a_1}\) in the sequence followed by the common ration \(\color{red}r\) raised to the power \(\color{red}{n-1}\)

Why????

because when you are going to find out the first term, you would plug in n=1 into this \(a_n=a_1*r^{n-1}\), so you got the first term \(a_1\) only, so you can try yourself any other term

Answer 9

So what will that mean?

Answer 10

For the problem

Answer 11

\[\Huge a_n=\color{blue}{a_1}*\color{lime}r^{n-1}\]

that means you plug in (4/9) instead of "the first term" \(a_1\) and 3 instead of the common ratio \(r\)

Answer 12

So how will that turn out?

Answer 13

It is your turn @Gamenerd123 now

Answer 14

Well, i mean like how do I do that, sorry.

Answer 15

Where is the first term?
and
where is the common ratio?

Answer 16

a1 is the first term and 3 is the common ratio, right?

Answer 17

a1=??

Answer 18

3?

Answer 19

if a1=3 and r=3, so 4/9=?????????????????

Answer 20

3

Answer 21

a_1=?
r=?

Answer 22

Its not 3?

Answer 23

no it is not..

\[\Huge a_1=\frac{ 4 }{ 9 }~~~,~~~ a_n=3*a_{n-1}\]
Do you remember that?

Answer 24

Yes

Answer 25

so?

Answer 26

Hmmm...

Answer 27

I really like who ask and stop the explanation when he sticks to something, even it looks so simple!

So don't be embarrassed if there is anything not clear for you

Answer 28

My minds so slow for math I'm sorry

Answer 29

It takes a while for me to get it

Answer 30

No no don't say that ...I am not agree with you!

Answer 31

I have another idea...

Answer 32

Please, do tell..

Answer 33

Ok..

Can you type the formula of the explicit rule for this sequence?

Answer 34

Not exactly...

Answer 35

I'm very dumb for formulas

Answer 36

got it?