Question

Help please!! I will fan and medal :>

Answer 1

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 2

So what do you have to do first?

Answer 3

Write his first 20 payments in sigma notation @iYuko

Answer 4

How do I write it though? That's honestly all I need help with ;v; @iYuko

Answer 5

\[\sum_{}^{}\]

Answer 6

http://www.mathsisfun.com/algebra/sigma-notation.html This explains

Answer 7

I see!! But how do I write this problem exactly? Like would it be 100*1.1^20 or...? @iYuko

Answer 8

Ummmm...@shifuyanli

Answer 9

I just know what the sigma means

Answer 10

aaa okay gotcha. Thank you though!! :^) @iYuko

Answer 11

he shouldn't go to college

Answer 12

@amistre64 is there by chance you can help me please?

Answer 13

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 14

whats his first payment?

Answer 15

$100! @amistre64

Answer 16

good, and his second payment is 1.1 times bigger, how do we represent that? whats is the 2nd payment in terms of the first?

Answer 17

100*1.1
100*1.1^2 ?? @amistre64

Answer 18

thats the 2nd and 3rd payments ok ... we need to define a rule for the nth payment

what would you suggest off hand?

Answer 19

would it be something like 100*1.1^19 ???

Answer 20

p1 = 100(1.1)^0
p2 = 100(1.1)^1
p3 = 100(1.1)^2
p4 = 100(1.1)^3

each payment number seems to have an exponent that is 1 less

pn = 100(1.1)^(n-1) seems to be a good rule

Answer 21

would that rule be considered his payments in the form of sigma notation? @amistre64

Answer 22

we would use that rule for sigma notation yes, since it defines 1 particular nth payment

Answer 23

brb

Answer 24

Thank you so much!! All I would have to do from here on is plug in 20 into our formula correct?? @amistre64

Answer 25

depends on how far weve gotten

right now we have the value of a particular payment, the sum is something else

100(1.1)^0 + 100(1.1)^1 + 100(1.1)^2 + ... + 100(1.1)^(n-1)

the sum of all the payments is the summation of the rule over the interval

\[\sum_{n=1}^{k}100(1.1)^{n-1}\]

by the way, what type of series is generated?

Answer 26

all n=20 does is tell us the amount of the 20th payment, it doesnt add them all up

if we know what type of series the paymets form, we can determine if we use the arithmatic formula of summation, or the geometric formula

Answer 27

We can use the geometric formula correct? @amistre64

Answer 28

geometric series ? @amistre64

Answer 29

yes, its a geometric sequence, so we can use the geometric series formula

\[S_n=a_1\frac{r^n-1}{r-1}\]
what are our parts?

Answer 30

a= 100
n = 20
r=1.1
so then it would be 100 (1.1^20 - 1 / 1.1 - 1) or around @amistre64

Answer 31

lets get your paranthesis right ...

100 (1.1^20 - 1)/(1.1 - 1)

100 (1.1^20 - 1)/(.1)

1000 (1.1)^20 - 1000

rnd if you want to i spose but what does this get us to start with?

Answer 32

yeah, we should rnd :)

5727.49994932560009201

Answer 33

why does the series converge?

Answer 34

I got that!! Thank you so much !! The series converges because it gets to a certain number right? @amistre64

Answer 35

yes, i was thinking more along the lines: a finite number of items is finite, and therefore the sum of a finite number of objects converges to a finite value.

Answer 36

ohh!! Ok thank you so much !!! you are such a livesaver ;v;!! @amistre64