Question

I NEED HELPPP!!
Tom has taken out a loan for college. He started paying off the loan with a first payment of $200. Each month he pays, he wants to pay back 1.2 times the amount he paid the month before. Explain to Tom how to represent his first 30 payments in sigma notation. Then explain how to find the sum of his first 30 payments.

Answer 1

@SolomonZelman

Answer 2

@Hero

Answer 3

@Hero please help it's really important

Answer 4

@phi can you please help?

Answer 5

@blues

Answer 6

@SolomonZelman @mathmate @Awesome781 @xGuardians @e.mccormick

Answer 7

@mathmate literally anything would help <3

Answer 8

@jim_thompson5910

Answer 9

What's the first payment?

Answer 10

$200

Answer 11

What's the next payment

Answer 12

and the 2nd payment ?

Answer 13

idk.. the second payment needs to be 1.2 times more than the first payment, soooo 240?

Answer 14

and the next?

Answer 15

288 but idk what sigma notation is :(

Answer 16

in general, do you know how to find the payment for any given month?

Answer 17

ummm I dont think so

Answer 18

Payment 1: 200
Payment 2: 240 = 200*1.2
Payment 3: 288 = 240*1.2

etc etc

Answer 19

notice how payment 3 can be written as

288 = 240*1.2 = (200*1.2)*1.2

Answer 20

Then (200*1.2)*1.2 = 200*(1.2*1.2) = 200*(1.2)^2

Answer 21

What's payment 4?

Answer 22

about 345. are you sure thats sigma notation?

Answer 23

we'll get there

Answer 24

288*1.2 = 345.6

We can write that as

345.6 = 288*1.2 = ((200*1.2)*1.2)*1.2 = 200*(1.2)^3

starting to see a pattern?

Answer 25

jim is explaining how to find each payment.
after you know that, we use a short-cut notation to write the answer
that will be the sigma notation.

But you have to know what the series is to write it down. Once you know it, the sigma part is relatively easy

Answer 26

oh okay. I think I get it now. Thank you all so much. I have one more problem can you guys please help? It's the last one

Answer 27

Bobby has been planting blackberries in a garden in his backyard. Bobby started with blackberries in one square foot of garden modeled by the function s(x) = 30. He is allowed to convert more of the garden, and each additional square foot he maintains allows his blackberry plants to produce more by a rate of a(x) = 1.3^(x-1). Explain to Bobby how to create an equation to predict the number of blackberries he can expect based on the number of square feet he maintains. Describe how to determine the number of blackberries he will grow with 20 square feet.

Answer 28

I know its long but I dont think its that complicated. please I really need help

Answer 29

before we move on, what's the nth term? What's the general formula to get any payment you want?

Answer 30

uh oh idk :o

Answer 31

like 200 * 1.2^2

Answer 32

that's a specific payment

I want the general payment for any month

Answer 33

I have no clue

Answer 34

First payment: 200 = 200*(1.2)^0
Second payment: 240 = 200*(1.2)^1
Third payment: 288 = 200*(1.2)^2
Fourth payment: 345.6 = 200*(1.2)^3

See the pattern?

Answer 35

yeah, but whats the general payment?

Answer 36

what's the pattern? or the basic template?

Answer 37

please just tell me lol Im like helen keller

Answer 38

wait so can you tell me how to represent these payments in sigma notation? I get everything else

Answer 39

What's the same each time? What's different?

look at the list again

First payment: 200 = 200*(1.2)^0
Second payment: 240 = 200*(1.2)^1
Third payment: 288 = 200*(1.2)^2
Fourth payment: 345.6 = 200*(1.2)^3

Answer 40

youre always multiplying by 200 and 1.2 and the exponent is different every time

Answer 41

good

Answer 42

so can you make a guess where the variable goes?

Answer 43

keep in mind that variable means "something that changes" (vary ---> vari-able, vary = change)

Answer 44

the exponent?

Answer 45

it's the only thing changing

Answer 46

good

Answer 47

so for the first payment

200 = 200*(1.2)^0

if n = 1 represents the first payment, then n-1 must be the exponent (to get 0)

Answer 48

That's why the general nth term is

\[\large 200(1.2)^{n-1}\]

Answer 49

oh okay. I get it. Can you tell me how to represent it in sigma notation now?

Answer 50

well you use the symbol \(\Large \sigma\)

this is the greek upper case letter sigma

Answer 51

Below the sigma, you tell where to start

You start at n = 1

Answer 52

Above the sigma, you tell where to stop. We stop at n = 30 (because we have 30 payments)

Answer 53

Off to the right, you write what you want to sum up, which is \[\large 200(1.2)^{n-1}\]

Answer 54

Put this all together to get

\[\large \sum_{n=1}^{30} 200(1.2)^{n-1}\]

Answer 55

That's the shorthand way to say "add up the payments from n = 1 to n = 30 from the nth term formula \(\Large 200(1.2)^{n-1}\) "

Answer 56

okay, I completely understand now. Thank you so much. Do you have any idea on that question I posted before?

Answer 57

oops I meant to write \(\Large \Sigma\) and NOT \(\Large \sigma\)

Answer 58

would the series be convergent or divergent? im stuck on that part..

Answer 59

@gennybaby99 notice how r = 1.2

since |r| < 1 is NOT true, this means the series does NOT converge

The series diverges. If you keep going forever, the sum grows and grows. It doesn't approach a single fixed number

Answer 60

thank you soo much