Question

guys i give medalsss!!!!! helpppp please
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 as 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, using complete sentences. Explain why this series is convergent or divergent.

Answer 1

@phi can you help me?

Answer 2

@Garadon_Shagan can you help me?

Answer 3

@ganeshie8 help

Answer 4

where exactly are you stuck ?

Answer 5

I dont know how to make the stigma

Answer 6

start by writing out first few payments,
then u wil get bored and see a pattern..

Answer 7

sigma notation is easy,
il help you with sigma notation... but first find out the pattern (nth term )

Answer 8

first payment = $200
secon payment = ?
third payment = ?

Answer 9

ok for the first payment he will pay 200 right, for the second one 240, the third one 288

Answer 10

how u got 240 for second payment ?

Answer 11

200*1.2

Answer 12

yes,
how u got 288 for 3rd payment ?

Answer 13

240*1.2

Answer 14

and 240 = 200*1.2, right ?

so can we write the 3rd payment as below :
(200*1.2)*1.2 = 200*1.2^2

?

Answer 15

In general, \(n\)th payment can be written as :
\[\large 200(1.2)^{n-1}\]

Answer 16

so if wee were to solve folve for payment #6 we just plug in 6 in the n?

Answer 17

exactly!

Answer 18

plugin n=1 and convince yourself if you really get ur first payment or not

Answer 19

okay but, you know the stigma has a lower and an upper part? what do i put there?

Answer 20

yeahhh i got 200 for the first one

Answer 21

good :) sigma notation is just a lazy way of writing sum of things

Answer 22

for example :

\[\large \sum \limits_{n=1}^4 n \]

is same as :

\[\large 1+2+3+4\]

Answer 23

\[\sum_{}^{?}\]

Answer 24

okay so for the upper it can be 30

Answer 25

and the lower n=1

Answer 26

yep ! you're trying to add up all the payments, right ?

Answer 27

yes

Answer 28

you're correct about the upper/lower index

Answer 29

:) so how do i know if it is divergent or convergent?

Answer 30

so whats ur sigma notation for the sum of first 30 payments ?

Answer 31

Sum of first 30 payments =
\[\large \sum \limits_{n=1}^{30} 200(1.2)^{n-1} \]


right ?

Answer 32

yes, let me calculate the answer

Answer 33

i got 39562.72

Answer 34

you need to use `geometric series` formula to find the sum

Answer 35

im getting 236376.3 :
http://www.wolframalpha.com/input/?i=+%5Csum+%5Climits_%7Bn%3D1%7D%5E%7B30%7D+%28200%281.2%29%5E%7Bn-1%7D%29

Answer 36

yes i got that answer too

Answer 37

so now i need to identify if it is convertgent or divergent

Answer 38

I think it is diverrgent because it has a common ratio right ?

Answer 39

do u mean, it has a common ratio > 1, so it is divergent ?

Answer 40

yes

Answer 41

then you would be correct :)

if |r| > 1, then the series is divergent

Answer 42

here r = 1.2, which is clearly greater than 1, so the series is divergent

Answer 43

okay i got it< thank you so much @ganeshie8 i really appreciate it

Answer 44

another way to think of it is :
you're adding atleast $200 every month, so the overall sum NEVER approaches any finite number, it always kepps GROWING forever...

Answer 45

oh okay :)