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 betore. Explain to Jo howto represent his first 20 payment n sigma notation.Then esplain how to tind the sum of his frst 20 pyments, using complete sentences. Esplain why this series is convergent or divergent
Since each term of the series is 1.1 times the previous one, the series is geometric. The generic term of a geometric series is [tex]a_{n}=a_{0}\cdot r^{(n-1)}[/tex]
The sum in sigma notation simply adds these terms. The leading factor of 100 can be factored out. [tex]sum=\displayform{100\cdot\sum\limits_{n=1}^{20}{1.1^{(n-1)}}}[/tex]
The sum can be found by adding the terms or by using the formula for the sum of a geometric series. In the latter case, we have [tex]sum=100\cdot\dfrac{1.1^{20}-1}{1.1-1}\approx5727.50[/tex]
The series is divergent because the common ratio of terms is greater than 1. (Of course, any finite number of terms will have a finite sum.)
