Question

What is the next number in the sequence: 9, 3, 1, 1/3,...
Answer is 1/9 but i am unsure how they got that answer, can someone show me how they got it? I get lost after getting 1/3, I thought it decreased by dividing all the numbers by 3 but that would mean that the answer should be .11111111 repeating not 1/9?

Answer 1

Hint :
\[0.111\ldots = \dfrac{1}{10} + \dfrac{1}{100}+\dfrac{1}{1000}+\cdots \]

Answer 2

Hahahaha I feel dumb
9/1 3/1 1/1 1/3 1/3
They are mirroring ?

Answer 3

That was suposed to be 1/9 the last one...

Answer 4

your teacher must be feeling patterny :)

Answer 5

did you get my earlier hint ?

Answer 6

Im studying for entrance exams. I want to go to collage.

Answer 7

I thought I did but no, I didnt.

Answer 8

Ohkay, lets take it step by step and slow

Answer 9

look at the number \[246.0\]

what does it represent ?

Answer 10

it is a shortcut for below lengthy expression :
\[2\times 10^2 + 4\times 10^1 + 6\times 10^0 + 0\times 10^{-1}\]

yes ?

Answer 11

Sorry I am usually not this slow. I would say 246/1 right? and I don't understand your last comment.

Answer 12

how do you "spell out" the number 246.0 ?

Answer 13

Two hundred fourty six

Answer 14

good, let me break that :

```
two hundred
fourty
six
```

Answer 15

same as

```
200
+
40
+
6
```

Answer 16

oh ok

Answer 17

same as

```
2x100
+
4x10
+
6x1
```

Answer 18

Now I understand your one comment.

Answer 19

246 represents :

2 hundreds
4 tens
6 ones

Answer 20

lets do a quick example

Answer 21

"spell out" the number 4789

Answer 22

Four thousand
seven hundred
eighty
nine

Answer 23

can you write it in numbers...

Answer 24

4000
700
80
6

Answer 25

```
4 x 1000
+
7 x 100
+
8 x 10
+
9 x 1
```

Answer 26

9

Answer 27

can we write it as :

```
4 x 10^3
+
7 x 10^2
+
8 x 10^1
+
9 x 1^0
```

?

Answer 28

I put a 6 not a 9. Continue sorry.

Answer 29

above is called "decimal expansion" of 4789

Answer 30

See if you can write the decimal expansion of 345

Answer 31

3x10^2
+
4x10^1
+
5x1^0

Answer 32

Excellent!
lets do one more example

Answer 33

write the decimal expansion of 328.6

Answer 34

3x10^2
+
2x10^1
+
8x10^0
+
.6X.1^0

Answer 35

wrong

2, 1, 0,

what comes next ?

Answer 36

.1

Answer 37

-1

Answer 38

-1, yes

Answer 39

so the decimal expansion of 328.6 is

```
3x10^2
+
2x10^1
+
8x10^0
+
6x10^(-1)
```

Answer 40

why would it be a negative instead of a decimal?

Answer 41

that is a very good question!
before answering that, could you "spell out" the number 328.6 ?

Answer 42

three hundred
twenty
eight
and
six tenths

Answer 43

very good

how do you "write out" six tenths ?

Answer 44

i mean,
how do you express six tenths using numbers ?

Answer 45

6x10^(-1)

Answer 46

how did u get -1 ?

Answer 47

you're correct btw..

Answer 48

Thats what you showed me when we expressed 328.6

Answer 49

I don't know how, that's what I was shown.

Answer 50

do you know six-tenths can also be written as 6/10? @Ashtorah13

Answer 51

Yes

Answer 52

\[\frac{6}{10}=\frac{6}{10^{1}}=6 \cdot \frac{1}{10^{1}}=6 \cdot 10^{-1}\]

Answer 53

Still don't know why its a negative and not a decimal.

Answer 54

the exponent is negative
not the number itself
the exponent is negative because it is a decimal

Answer 55

Ok so its just a different way to show the decimal?

Answer 56

examples:
\[.6=\frac{6}{10} \text{ or } 6 \cdot 10^{-1} \\ .06 =\frac{6}{100}=\frac{6}{10^2} \text{ or } 6 \cdot 10^{-2} \\ .006 =\frac{6}{1000}=\frac{6}{10^3} \text{ or } 6 \cdot 10^{-3}\]

Answer 57

OH ok so it shows where the decimal will go!

Answer 58

so like 6X10^(-5) would be .00006

Answer 59

yes like 6*10^2 would be 06.00 move the decimal over 2 spaces to the right so you have 0600. or just 600
but 6*10^(-2) would be 06.00 move the decimal over 2 spaces to left so you have .0600 or just .06

Answer 60

yes

Answer 61

Bingo Thank you. But I still don't understand my original question...

Answer 62

what is the original question

Answer 63

are you talking about the next number in that sequence you first mentioned?

Answer 64

Yes

Answer 65

I feel what I just learned ties into it but I am unsure

Answer 66

do you know what common ratio means?

Answer 67

No? (I may, I just don't remember.)

Answer 68

oh actually you did see the common ratio earlier

what you meant to say earlier is that the patterned looked like all you had to do was take term and divided by 3 to get next term right?

Answer 69

Ya

Answer 70

another way to say divide by 3 is to say multiply by 1/3

Answer 71

9*1/3=3
3*1/3=1
1*1/3=1/3
1/3*1/3=?

Answer 72

It makes perfect since now!

Answer 73

don't whip out your calculator

Answer 74

just multiply straight across :)

Answer 75

and you are right .11111111111111111111111111111111111111111111111111111..... is 1/9 :)

Answer 76

most calculators won't tell you that though
a bad side effect of some calculators :p

Answer 77

common numerator dont need to, :3
I knew it was 1/9 just wasnt sure how they got it.

and oh my, how the heck would you simplify .1111111111 into 1/9?

Answer 78

Is there a way to give you both medals?

Answer 79

we can even prove 1/9 is .1111111111111111111111111111111111111111(repeating)


\[x=.1111111111111111111.... \\ \text{ multiply both sides by 10 } \\ 10x=1.111111111111111111111................... \\ \text{ now take equation 2 and subtract from \it equation 1 } \\ 10x-x=1.111111111111111111111...........-.1111111111111111111.... \\ 9x=1.00000000000000000..... \\ 9x=1 \\ \text{ divide both sides by 9 } \\ x=\frac{1}{9}\]

Answer 80

therefore .111111(repeating)=1/9

Answer 81

another example:

Say we wanted to figure out what fraction .534534534534.... equals


let x=.534534534534....
multiply both sides by 1000 repetition begins after 1000th place
\[1000x=534.534534534......\]

now we subtract 2nd from 1st equation again

\[999x=534.00000000000000000000.... \\ \text{ so } x=\frac{534}{999}\]

check your calculator if you want to

Answer 82

you will see 534/999 is .534(repeating)

Answer 83

now we subtract 1st from 2nd equation *

Answer 84

Is anything repeating over 9? Say you have .7777777
would that mean it would be 777/999?

Answer 85

let's find out

let x=.7777777777777(repeating)
repetition starts after 10th's place so we multiply both sides by 10
10x=7.77777777777(repeating)

subtract 1st from 2nd we get
10x-x=7
9x=7

solve for x by dividing both sides by 9

x=7/9

and you said 777/999 which still works since 777/999 can be reduced to 7/9


but yeah you are right (you just have to put the right amount of 9's depending on the repetition)
sometimes you can reduce the answer just like what you found with 777/999

Answer 86

another example for fun

\[x=.3401340134013401.... \\ \text{ here we multiply both sides by } 10000 \\ \text{ since repetition begins after } 10000th \text{ place} \\ \\ 10000x=3401.34013401...... \\ \text{ subtract 1st from 2nd } \\ 10000x-x=3401 \\ 9999x=3401 \\ x=\frac{3401}{9999}\]

Answer 87

10x=7.77777777777 - x=.7777777777777
(let me see if I can explainthis right)
I was uncertine how you kept getting 9 from 10 after subtracting. Is it when you subtract 10x from x you have to put a 1 infront of the x? Making it 10x- 1x? So you get 9x?

Answer 88

that is correct x is 1x

Answer 89

Oh my this is making more and more since. My algebra teacher wasn't vary good.