Question

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

hint: this is a geometric sequence with

a = 200 being the first term
r = 1.2 being the common ratio

Answer 2

What was the formula for a geometric sequence again? I'm doing a lot of essay questions that need to be done in 20 minutes lol

Answer 3

The nth term of a geometric sequence is

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

Answer 4

And n would be the number of payments correct?

Answer 5

n is the nth payment

so say n = 3, that would represent the 3rd payment. To find the actual 3rd payment, plug n = 3 into the nth term expression I wrote above

Answer 6

a = 200
r = 1.2

those two values above are going to stay fixed like that

Answer 7

Okay so it would be \[200(1.2)^{3-1}=344.6\]

Answer 8

you should get this

\[\Large 200(1.2)^{3-1}=288\]

so that means payment n = 3 is $288

Answer 9

the other values of n are calculated in a similar way

Answer 10

My calculator sucks haha. I got it now. If we substitute 4 for 3 we'd get \[200(1.2)^{4-1}=414.72?\] or am I doing it wrong?

Answer 11

it's incorrect

Answer 12

the "4-1" turns into "3"

Answer 13

if you are typing into a calculator, make sure to use parenthesis

200*(1.2)^(4-1)

Answer 14

http://web2.0calc.com/#200*(1.2)^(4-1)

Answer 15

Would it be 345.6?

Answer 16

yes

Answer 17

so imagine this happening all the way from n = 1 to n = 30

we could write out every single term, but that's very tedious and not necessary with proper math notation

we can use sigma notation to represent a shortcut which essentially says "add up a bunch of these terms"



that funky looking E is the capital greek letter sigma

Answer 18

it means "add up a bunch of terms"