Larry has taken out a loan for college. He started paying off the loan with a first payment of $150. Each month he pays, he wants to pay back 1.3 times as the amount he paid the month before. Explain to Larry how to represent his first 15 payments in sigma notation. Then explain how to find the sum of his first 15 payments, using complete sentences. Explain why this series is convergent or divergent.

Question

amistre64 · Answer

so whats your explanation so far?

anonymous · Answer

so far i have the equation a1-a1r^n/1-r which i think would accompany this question but I'm not sure

anonymous · Answer

so far, I'm just doing the math, trying to figure out what the formula may be

amistre64 · Answer

that at least has something to do with a geometric series :)

anonymous · Answer

lol A for effort

amistre64 · Answer

lets start out with a payment: P

month1 = P
month2 = (P) * k
month3 = (P * k) * k
month4 = ((P * k) * k) * k



simplifing this stuff we see that

month1 = P
month2 = Pk
month3 = Pk^2
month4 = Pk^3
....
month n = Pk^(n-1)

agreed?

anonymous · Answer

YES, completely

anonymous · Answer

so now, all i do is substitute my values?

amistre64 · Answer

well, then thats how the sequence is generates

the sum of payments is just the sum from months 1 to n;  and yes you would want to substitute your values into this generalization

amistre64 · Answer

$$\sum_{n=1}^{j}Pk^{n-1}$$
this of course can be written in several ways but is a good form as is

amistre64 · Answer

since j=15,  would you agree that the sum of 15 numbers is somehting that produces a fininte value?

anonymous · Answer

no of coarse not

amistre64 · Answer

for example; spose you take out all the change from your piggy bank ... there may be hundreds of coins, even thousands .... or maybe just 2 coins

but if we count the value of all those coins, we come to some knowable value right?

anonymous · Answer

yeah of coarse

amistre64 · Answer

then we can count 15 payments and reach some known value ... it converges to some number

amistre64 · Answer

you can use the formula you presented at first to determine that value, but in terms of what ive been using:$$S_j=P\frac{1-k^j}{1-k}$$

anonymous · Answer

exactly, it has to reach a certain point, it doest go on for infinity, right?

amistre64 · Answer

right :)

anonymous · Answer

oh alright, i think i get it, so this is the sum? great!! :D

amistre64 · Answer

with any luck :)

anonymous · Answer

so for 15 months, it would be 25092?

amistre64 · Answer

thats correct

amistre64 · Answer

i have to go pick up my kids, so have fun with this :)

anonymous · Answer

Thank yo so much!! Have a blessed day with your kids <3