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.

Question

anonymous · Answer

I'm rather confused by this math question and what to do here...

tkhunny · Answer

100
100*1.1
100*1.1^2
100*1.1^3
...
100*1.1^19

There they are.  You do the sigma notation.

anonymous · Answer

How?

anonymous · Answer

Nevermind, I got it. Thank you!

tkhunny · Answer

Did you succeed in adding them up?

anonymous · Answer

for the most part, i did.

tkhunny · Answer

100 + 100*1.1 + 100*1.1^2 + 100*1.1^3 + ... + 100*1.1^19
= 100(1 + 1.1 + 1.1^2 + 1.1^3 + ... + 1.1^19)

= $100\dfrac{1-1.1^{20}}{1-1.1} = 100\cdot\dfrac{5.7275}{0.1} = 5727.50$

You must be familiar with Geometric Series.  Very helpful skills.

anonymous · Answer

@Hero is this series convergent or divergent?

hero · Answer

If the series is convergent, then it approaches a certain number. If it is divergent, then it doesn't.

anonymous · Answer

ok @Hero thank you so much