Question

John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. Explain to John how to represent his first 20 payments in sigma notation. Then explain how to find the sum of his first 20 payments, using complete sentences. Explain why this series is convergent or divergent.

Answer 1

@Kainui

Answer 2

@Safa102 can you help me plz?

Answer 3

ok so i don't know what sigma notation is but i'm pretty sure what u have to do is, do 20 divided by 1.1 than get your answer than subtract that by 20..i'm not sure but this is what i got. i hope this helps :)

Answer 4

Okay thanks :)

Answer 5

np! :)

Answer 6

@Kainui plz help?

Answer 7

@hartnn can you help me plz?

Answer 8

@jim_thompson5910

Answer 9

This is a geometric series, so you'll use the nth term of the geometric sequence formula, which is

\[\Large a_n = a*(r)^{n-1}\]

look familiar?

Answer 10

Yes

Answer 11

what would the value of 'a' and 'r' be?

Answer 12

A = 20
R = 1.1?

Answer 13

a = 20 is not correct

Answer 14

So would a = 100?

Answer 15

yes that's the starting amount

Answer 16

So it will be 100 * (1.1)^n-1
Where n = 20

Answer 17

you're close but not quite there

Answer 18

\[\Large \sum_{k=0}^{n}\left[a*(r)^{n-1}\right] = a*\frac{1-r^n}{1-r}\]
\[\Large \sum_{k=0}^{n}\left[100*(1.1)^{n-1}\right] = 100*\frac{1-(1.1)^n}{1-1.1}\]

Answer 19

Sorry I don't get this. Can you please explain?

Answer 20

sorry I made a typo, let me fix

Answer 21

\[\Large \sum_{k=0}^{n}\left[a*(r)^{k-1}\right] = a*\frac{1-r^n}{1-r}\]
\[\Large \sum_{k=0}^{n}\left[100*(1.1)^{k-1}\right] = 100*\frac{1-(1.1)^n}{1-1.1}\]

Answer 22

are you familiar with sigma notation? and what it represents?

Answer 23

Not really

Answer 24

sigma is used for summations

so adding numbers up but instead of saying something like

1+2+3+4+5+6+7+8+9+10

which is a lot to write down, we can condense all that into

\[\Large \sum_{k=1}^{10}k\]

Answer 25

so when they're asking for sigma notation for the payments, they are asking you to add up all of the payments. You could take the long way of adding up each one by one or you can use the shortcut formula

Answer 26

So since he payed 100 first and wants to pay 1.1 times the amount in his first 20 payments it will be 1.1*100

Answer 27

does my example where I wrote out `1+2+3+4+5+6+7+8+9+10` make sense?

Answer 28

Yes

Answer 29

the first payment is $100, this is given

what is the next payment equal to?

Answer 30

110

Answer 31

yes

Answer 32

how about the next payment?

Answer 33

121

Answer 34

yes, so we're going to have

100+110+121+...

all the way up to 20 payments

Answer 35

:O your still at it...wow

Answer 36

now you could find all 20 values and add them all up, or you can take the shortcut which is what I recommend

so....

\[\Large a*\frac{1-r^n}{1-r} = 100*\frac{1-(1.1)^{20}}{1-1.1} = ??\]

a = 100 is the first payment
r = 1.1 is the multiplier
n = 20 is the number of payments

Answer 37

5727.49

Answer 38

I'm getting `5,727.4999493256` which rounds to `5,727.50`

Answer 39

Ok thanks!

Answer 40

I get it now :)

Answer 41

you're welcome