Question

I'm so confused on this, I'm not sure what I'm supposed to use for any of this, I don't know the common denominator or anything.
Find the 6th term of the sequence with t1 = -4 and tn = 5tn-1.

http://imgur.com/BaADfyq

Answer 1

start with \(-4\), and keep multiplying by \(5\) to get the next term.

Answer 2

So T6 is -62,500?

Answer 3

first term, \(\large t_1 ~=~-4\)

second term, \(\large t_2 ~=~-4*5 = -20\)

third term, \(\large t_3 ~=~-20*5 = -100\)

keep going till the 6th term

Answer 4

-62,500 is wrong

Answer 5

Sorry, I did too many of them. -2500.

Answer 6

show me the work instead

Answer 7

T4 = -100 x 5 = -500
T5 = -500 x 5 = -2500
T6 = -2500 x 5 = -12500

Answer 8

Good!

Answer 9

And then how do I go about finding the sum of an infinite series?

Answer 10

look at the terms :
\[-4,~~-20,~~-100,~~-500,~~-2500,~~ \ldots\]

what do you notice ? do they seem to converge to some number ?

Answer 11

You could go on for ever without getting a sum.

Answer 12

Yes, so we say the "sequence doesn't converge" and the infinite sum doesn't exist

Answer 13

So for an equation like this - Find the sum of the infinite series 3 + 1.2 + 0.48 + 0.192 + ...if it exists.


All I would do is use the formula to find the equation of the infinite series?

Answer 14

what kind of series is it, geometric/arithmetic ?

Answer 15

Geometric?

Answer 16

there is a nice criterion for testing convergence of geometric series, you simply look at the common ratio

Answer 17

whats the common ratio of given series ?

Answer 18

The equation is tn/(tn-1)

Answer 19

Common ratio is 2.5

Answer 20

yes work it

common ratio = (next term)/(present term)

Answer 21

looks you have worked it in reverse : (present term)/(next term)

Answer 22

try again

Answer 23

.4?

Answer 24

yes common ratio = 0.4 which is between -1 and 1
so the given series conveges

Answer 25

A geometric series converges if the common ratio is between -1 and 1

Answer 26

use infinite sum formula to find the sum

Answer 27

do you have the formula wid u ?

Answer 28

S(infinite) = 3/t-.4

Answer 29

I have it in my notes

Answer 30

\[\large S_{\infty} ~=~\dfrac{t_1}{1-r}\]

Answer 31

\[\large S_{\infty} ~=~\dfrac{3}{1-0.4}\]

Answer 32

simplify

Answer 33

5?

Answer 34

Yes!

Answer 35

I have three more if you have time?

Answer 36

okay il try, post

Answer 37

http://gyazo.com/f47f214fcdb2adb56c90ca6794e3f48d

Answer 38

Firs test if the series converges by finding the common ratio

Answer 39

whats the common ratio ?

Answer 40

1.33?

Answer 41

Yes, which is "not" in between -1 and 1
so the series does not converge.

Answer 42

we say the sum does not exist

Answer 43

Okay so the answer would be "No solution"

Answer 44

As you can see, the common ratio decides whether an infinite converges to some number, or if it diverges

Answer 45

Answer would be "does not converge"

Answer 46

If it converges then it has a sum, if it does not converge there is not really a solution?

Answer 47

yes, do you see why the common ratio of 1.33 gives a diverging sum ?

Answer 48

Because it's larger then 1? and not between that -1 and 1

Answer 49

Yes, when you multiply the first term by 1.33, you get a bigger next term; the terms keep growing and the sum wont reach a specific number

Answer 50

So for this problem - http://gyazo.com/15138c898cca305ec43a79e298d55821

The answer would be .498

Answer 51

how did u get 0.498 ?

Answer 52

im asking because im getting a more nice looking number : 0.5

Answer 53

I didn't round.

Answer 54

me neither
could you show ur work please

Answer 55

\[S \infty = \frac{ 1 }{ 3 }\div 1-.33\]

Answer 56

Ahh okay, you're rounding 1/3 to 0.33

Answer 57

i worked it like this :
\[\large S_{\infty}~=~\dfrac{1/3}{1-1/3} = \dfrac{1}{3-1}=\dfrac{1}{2}=0.5\]

Answer 58

I'm not sure why I changed from a fraction to a decimal. Fractions are more precise aren't they?

Answer 59

Yep

Fractions are exact
decimals are not so exact when you get a repeating decimal

Answer 60

1/2 = 0.5

both are exact

Answer 61

1/3 = 0.33333333333333333333333333...

clearly the decimal wont be exact because you can't write out infinitely many 3's

Answer 62

Okay, last one - http://gyazo.com/50d340d91d94093d361677d900fa46ee

Answer is 1.

Answer 63

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Answer 64

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Answer 65

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Answer 66

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Answer 67

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Answer 68

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Answer 69

Finding the sum of an geometric Find S12 for the series 1 + 2 + 4 + 8 +... like this is 49,140

Answer 70

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Answer 71

how did u get 49.140 ?

Answer 72

Sorry it should be 8,190.


2((1-2^12)/(1-2)) = 8190